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Silicate minerals make up the majority of the earth's crust and account for almost 92 percent of the total. Silicate sheets, often known as silicate networks, are characterised as definite connectivity parallel designs. A key idea in studying different generalised classes of graphs in terms of planarity is the face of the graph. It plays a significant role in the embedding of graphs as well. Face index is a recently created parameter that is based on the data from a graph's faces. The current draft is utilizing a newly established face index, to study different silicate networks. It consists of a generalized chain of silicate, silicate sheet, silicate network, carbon sheet, polyhedron generalized sheet, and also triangular honeycomb network. This study will help to understand the structural properties of chemical networks because the face index is more generalized than vertex degree based topological descriptors.
Citation: Ricai Luo, Khadija Dawood, Muhammad Kamran Jamil, Muhammad Azeem. Some new results on the face index of certain polycyclic chemical networks[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8031-8048. doi: 10.3934/mbe.2023348
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Silicate minerals make up the majority of the earth's crust and account for almost 92 percent of the total. Silicate sheets, often known as silicate networks, are characterised as definite connectivity parallel designs. A key idea in studying different generalised classes of graphs in terms of planarity is the face of the graph. It plays a significant role in the embedding of graphs as well. Face index is a recently created parameter that is based on the data from a graph's faces. The current draft is utilizing a newly established face index, to study different silicate networks. It consists of a generalized chain of silicate, silicate sheet, silicate network, carbon sheet, polyhedron generalized sheet, and also triangular honeycomb network. This study will help to understand the structural properties of chemical networks because the face index is more generalized than vertex degree based topological descriptors.
In this paper we study systems of linear hyperbolic equations on a bounded interval, say,
∂t(υϖ)=(−C+00C−)∂x(υϖ),0<x<1,t>0, | (1a) |
Ξ(υ(0,t),ϖ(1,t),υ(1,t),ϖ(0,t))T=0,t>0, | (1b) |
υ(x,0)=˚υ(x),ϖ(x,0)=˚ϖ(x)0<x<1, | (1c) |
where
\begin{equation} \boldsymbol{\Xi}_{out} (\boldsymbol{{ \upsilon}}(0, t), \boldsymbol{{ \varpi}}(1, t))^T + \boldsymbol{\Xi}_{in}(\boldsymbol{{ \upsilon}}(1, t), \boldsymbol{{ \varpi}}(0, t))^T = 0, \quad t > 0. \end{equation} | (2) |
An important class of such problems arises from dynamical systems on metric graphs. Let
\begin{equation} {\partial}_t \boldsymbol{p}^j+ \mathcal{M}^j{\partial}_x\boldsymbol{p}^j = 0, \quad 0 < x < 1, \; t > 0, \; 1\leq j\leq m, \end{equation} | (3) |
where
Such problems have been a subject of intensive research, both from the dynamics on graphs, [1,10,5,4,11,17,19], and the 1-D hyperbolic systems, [7,21,15,14], points of view. However, there is hardly any overlap, as there seems to be little interest in the network interpretation of the results in the latter, while in the former the conditions on the Riemann invariants seem to be "difficult to adapt to the case of a network", [11,Section 3].
The main aim of this paper, as well as of the preceding one [2] is to bring together these two approaches. In [2] we have provided explicit formulae allowing for a systematic conversion of Kirchhoff's type network transmission conditions to (1b) in such a way that the resulting system(1) is well-posed. We also gave a proof of the well-posedness on any
To briefly describe the content of the paper, we observe that if the matrix
\widehat {\left(\widehat{\boldsymbol{\Xi}_{out}}\right)^T\widehat{\boldsymbol{\Xi}_{in}}} |
is the adjacency matrix of a line graph (where for a matrix
The main idea of this paper is similar to that of [3]. However, [3] dealt with first order problems with (2) solved with respect to the outgoing data. Here, we do not make this assumption and, while(1) technically is one-dimensional, having reconstructed
The paper is organized as follows. In Section 2 we briefly recall the notation and relevant results from [2]. Section 3 contains the main result of the paper. In Appendix we recall basic results on line graphs in the interpretation suitable for the considerations of the paper.
We consider a network represented by a finite, connected and simple (without loops and multiple edges) metric graph
\mathcal{F}^j = \left(\begin{array}{cc} f^j_{+, 1}&f^j_{-, 1}\\ f^j_{+, 2}&f^j_{-, 2} \end{array}\right), |
the diagonalizing matrix on each edge. The Riemann invariants
\begin{equation} \boldsymbol{u}^j = (\mathcal{F}^{j})^{-1} \boldsymbol{p}^j\quad\text{and}\quad \boldsymbol{p}^j = \binom{f^j_{+, 1} u^j_1 + f^j_{-, 1}u^j_2}{f^j_{+, 2}u^j_1 + f^j_{-, 2}u^j_2}. \end{equation} | (4) |
Then, we diagonalize (3) and, discarding lower order terms, we consider
\begin{equation} {\partial}_t\boldsymbol{u}^j = \mathcal{L}^j{\partial}_x\boldsymbol{{u}}^j = \left(\begin{array}{cc}- \lambda^j_+&0\\0&- \lambda^j_- \end{array}\right){\partial}_x\boldsymbol{{u}}^j, \end{equation} | (5) |
for each
Remark 1. We refer an interested reader to [7,Section 1.1] for a detailed construction of the Riemann invariants for a general
The most general linear local boundary conditions at
\begin{equation} \boldsymbol{\Phi}_{ \mathbf{v}} \boldsymbol{p}( \mathbf{v}) = 0, \end{equation} | (6) |
where
\begin{equation} \boldsymbol{\Phi}_{ \mathbf{v}} : = \left(\begin{array}{ccccc}\phi^{j_1}_{ \mathbf{v}, 1}& \varphi^{j_1}_{ \mathbf{v}, 1}&\ldots&\phi^{j_{|J_{ \mathbf{v}}|}}_{ \mathbf{v}, 1}& \varphi^{j_{|J_{ \mathbf{v}}|}}_{ \mathbf{v}, 1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \phi^{j_1}_{ \mathbf{v}, k_{ \mathbf{v}}}& \varphi^{j_1}_{ \mathbf{v}, k_{ \mathbf{v}}}&\ldots&\phi^{j_{|J_{ \mathbf{v}}|}}_{ \mathbf{v}, k_{ \mathbf{v}}}& \varphi^{j_{|J_{ \mathbf{v}}|}}_{ \mathbf{v}, k_{ \mathbf{v}}}\end{array}\right), \ \end{equation} | (7) |
where
\begin{equation} \boldsymbol{\Psi}_{ \mathbf{v}} \boldsymbol{u}( \mathbf{v}) : = \boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}( \mathbf{v})\boldsymbol{u}( \mathbf{v}) = 0. \end{equation} | (8) |
For Riemann invariants, we can define their outgoing values at
Definition 2.1. Let
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Denote by
\begin{equation} k_{ \mathbf{v}} : = \sum\limits_{j\in J_{ \mathbf{v}}} (2(1-\alpha_j)l_j( \mathbf{v}) +\alpha_j ). \end{equation} | (9) |
Definition 2.2. We say that
● a sink if either
● a source if either
● a transient (or internal) vertex if it is neither a source nor a sink.
We denote the sets of sources, sinks and transient vertices by
We observe that if
A typical example of (8) is Kirchhoff's law that requires that the total inflow rate into a vertex must equal the total outflow rate from it. Its precise formulation depends on the context, we refer to [8,Chapter 18] for a detailed description in the context of flows in networks. Since it provides only one equation, in general it is not sufficient to ensure the well-posedness of the problem. So, we introduce the following definition.
Definition 2.3. We say that
To realize the requirement that the outgoing values should be determined by the incoming ones, we have to analyze the structure of
\begin{equation} \{1, \ldots, m\} = : J_1\cup J_2\cup J_0, \end{equation} | (10) |
where
J_{ \mathbf{v}} : = J_{ \mathbf{v}, 1}\cup J_{ \mathbf{v}, 2}\cup J_{ \mathbf{v}, 0}. |
We also consider another partition
\begin{equation} k_{ \mathbf{v}} = \sum\limits_{j\in J^0_{ \mathbf{v}}} \alpha_j + \sum\limits_{j\in J^1_{ \mathbf{v}}} (2-\alpha_j) = |J_{ \mathbf{v}, 1}|+ 2(|J^0_{ \mathbf{v}}\cap J_{ \mathbf{v}, 2}| + |J^1_{ \mathbf{v}}\cap J_{ \mathbf{v}, 0}|). \end{equation} | (11) |
Then, by [2,Lemma 3.6],
We introduce the block diagonal matrix
\begin{equation} \mathcal{{\tilde{{F}}}}_{out}( \mathbf{v}) = {\rm diag}\{\mathcal{{\tilde{{F}}}}_{out}^j( \mathbf{v})\}_{j \in J_{ \mathbf{v}}}, \end{equation} | (12) |
where
\mathcal{{\tilde{F}}}_{out}^j( \mathbf{v}) = \left\{\begin{array} {ccc} \left(\begin{array}{cc}0&0\\0&0\end{array}\right)&\text{if} &j\in (J_{ \mathbf{v}, 0} \cap J_{ \mathbf{v}}^0)\cup (J_{ \mathbf{v}, 2}\cap J_{ \mathbf{v}}^1), \\ \left(\begin{array}{cc}f^j_{+, 1}(l_j( \mathbf{v}))&f^j_{-, 1}(l_j( \mathbf{v}))\\f^j_{+, 2}(l_j( \mathbf{v}))&f^j_{-, 2}(l_j( \mathbf{v}))\end{array}\right)&\text{if}& j\in (J_{ \mathbf{v}, 0} \cap J_{ \mathbf{v}}^1)\cup (J_{ \mathbf{v}, 2}\cap J_{ \mathbf{v}}^0), \\ \left(\begin{array}{cc}f^j_{+, 1}(0)&0\\f^j_{+, 2}(0)&0\end{array}\right)&\text{if} &j\in J_{ \mathbf{v}, 1} \cap J_{ \mathbf{v}}^0, \\ \left(\begin{array}{cc}0&f^j_{-, 1}(1)\\0&f^j_{-, 2}(1)\end{array}\right)&\text{if}& j\in J_{ \mathbf{v}, 1} \cap J_{ \mathbf{v}}^1. \end{array} \right. |
Further, by
In a similar way, we extract from
\widetilde{\boldsymbol{u}}^j_{out}( \mathbf{v}) = \left\{\begin{array} {ccc} (0, 0)^T&\text{if} &j\in (J_{ \mathbf{v}, 0} \cap J_{ \mathbf{v}}^0)\cup (J_{ \mathbf{v}, 2}\cap J_{ \mathbf{v}}^1), \\ (u^j_1(l_j( \mathbf{v})), u^j_2(l_j( \mathbf{v})))^T&\text{if}& j\in (J_{ \mathbf{v}, 0} \cap J_{ \mathbf{v}}^1)\cup (J_{ \mathbf{v}, 2}\cap J_{ \mathbf{v}}^0), \\ (u^j_1(0), 0)^T&\text{if} &j\in J_{ \mathbf{v}, 1} \cap J_{ \mathbf{v}}^0, \\ (0, u^j_2(1))^T&\text{if}& j\in J_{ \mathbf{v}, 1} \cap J_{ \mathbf{v}}^1, \end{array} \right. |
and
Proposition 1. [2,Proposition 3.8] The boundary system (8) at
\begin{equation} \boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}_{out}( \mathbf{v}) \boldsymbol{u}_{out}( \mathbf{v}) + \boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}_{in}( \mathbf{v})\boldsymbol{u}_{in}( \mathbf{v}) = 0 \end{equation} | (13) |
and hence it uniquely determines the outgoing values of
\begin{equation} \boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}_{out}( \mathbf{v})\quad \mathit{\text{is nonsingular}}. \end{equation} | (14) |
In this case,
\begin{equation} \boldsymbol{u}_{out}( \mathbf{v}) = - (\boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}_{out}( \mathbf{v}))^{-1} \boldsymbol{\Phi}_{ \mathbf{v}} \mathcal{F}_{in}( \mathbf{v})\boldsymbol{u}_{in}( \mathbf{v}). \end{equation} | (15) |
To pass from (3) with Kirchhoff's boundary conditions at each vertex
\begin{equation} \boldsymbol{\Psi}' \gamma \boldsymbol{u} = 0. \end{equation} | (16) |
We note that the function values that are incoming at
\begin{equation} \boldsymbol{\Psi}^{out} \gamma \boldsymbol{u}_{out} + \boldsymbol{\Psi}^{in}\gamma\boldsymbol{u}_{in} = 0, \end{equation} | (17) |
where
Using the adopted parametrization and the formalism of Definition 2.1, we only need to distinguish between functions describing the flow from
\begin{equation} \begin{split} \boldsymbol{{\upsilon}} &: = \left((u^j_1)_{j\in J_1\cup J_2}, (u^j_2)_{j\in J_2}\right) = ( \upsilon_j)_{j\in J^+}, \\ \boldsymbol{{ \varpi}}& : = \left((u^j_1)_{j\in J_0}, (u^j_2)_{j\in J_1\cup J_0}\right) = ( \varpi_j)_{j\in J^-}, \end{split} \end{equation} | (18) |
where
In this way, we converted
Using this construction, the second order hyperbolic problem (3), (17) was transformed into first order system (1) with (17) written in the form (2). However, it is clear that (1) can be formulated with an arbitrary matrix
how to characterize matrices \boldsymbol{\Xi} that arise from \boldsymbol{\Psi} so that (1) describes a network dynamics?
For a graph
\boldsymbol{{\Psi}}_{ \mathbf{v}}^{out} = (\psi_{ \mathbf{v}, i}^j)_{1\leq i\leq k_{ \mathbf{v}}, j\in \boldsymbol{J}_{ \mathbf{v}}^-}, \qquad \boldsymbol{{\Psi}}_{ \mathbf{v}}^{in} = (\psi_{ \mathbf{v}, i}^j)_{1\leq i\leq k_{ \mathbf{v}}, j\in \boldsymbol{J}_{ \mathbf{v}}^+}. |
Since no outgoing value should be missing, we adopt the following
Assumption 1. No column or row of
These matrices provide some insight into how the arcs are connected by the flow which is an additional feature, superimposed on the geometric structure of the incoming and outgoing arcs at the vertex. In principle, these two structures do not have to be the same, that is, it may happen that the substance flowing from
Definition 3.1. Let
Using this idea, we define a connectivity matrix
\mathsf{c}_{ \mathbf{v}, lj} = \left\{\begin{array}{lcl} 1&\text{if}& \boldsymbol{{\varepsilon}}^j\;\text{flow connects to}\; \boldsymbol{{\varepsilon}}^l, \\ 0&&\text{otherwise}. \end{array} \right. |
Remark 2. We observe that
● the above definition implies that for
●
For an arbitrary matrix
Lemma 3.2. If
\begin{equation} \mathsf{C}_{ \mathbf{v}} = \widehat{\left(\widehat{\boldsymbol{{\Psi}}^{out}_{ \mathbf{v}}}\right)^T \widehat{\boldsymbol{{\Psi}}^{in}_{ \mathbf{v}}}}. \end{equation} | (19) |
Proof. Denote
\sum\limits_{r = 1}^{k_{ \mathbf{v}}} \hat{\psi}_{ \mathbf{v}, r}^i\hat{\psi}^j_{ \mathbf{v}, r} \neq 0. |
This occurs if and only if there is
Let
Definition 3.3. Let
As before, we construct a connectivity matrix
\begin{equation} \mathsf{c}_{ \mathbf{v}, ij} = \left\{\begin{array}{lcl} 1&\text{if}& \boldsymbol{{\varepsilon}}^j\;\text{and}\; \boldsymbol{{\varepsilon}}^i\;\text{are flow connected}, \\ 0&&\text{otherwise}. \end{array} \right. \end{equation} | (20) |
Note that, contrary to an internal vertex, here the connectivity matrix is symmetric. We also do not stipulate that
Lemma 3.4. If
\begin{equation} \mathsf{C}_{ \mathbf{v}} = \widehat{\left(\widehat{\boldsymbol{{\Psi}}^{out}_{ \mathbf{v}}}\right)^T \widehat{\boldsymbol{{\Psi}}^{out}_{ \mathbf{v}}}}. \end{equation} | (21) |
Proof. As before, let
\sum\limits_{r = 1}^{k_{ \mathbf{v}}} \hat{\psi}_{ \mathbf{v}, r}^i\hat{\psi}^j_{ \mathbf{v}, r} \neq 0. |
Certainly, by Assumption 1,
We adopt an assumption that the structure of flow connectivity is the same as of the geometry at the vertex. Thus, if
Assumption 2. For all
\mathsf{C}_{ \mathbf{v}} = \boldsymbol{1}_{ \mathbf{v}} = \left(\begin{array}{cccc}1&1&\ldots&1\\ \vdots&\vdots&\vdots&\vdots\\ 1&1&\ldots&1 \end{array}\right). |
We observe that the dimension of
If
Assumption 3. For all
Remark 3. Assumption 3 is weaker than requiring each two arcs from
Proposition 2. Let
\begin{equation} \boldsymbol{\Psi}_{ \mathbf{v}} \boldsymbol{u}( \mathbf{v}) = 0, \end{equation} | (22) |
contains a Kirchhoff's condition
\begin{equation} \sum\limits_{j \in J_{ \mathbf{v}}} (\psi^j_{ \mathbf{v}, r} u^j_1( \mathbf{v}) + \psi^j_{ \mathbf{v}, r} u^j_2( \mathbf{v})) = 0, \end{equation} | (23) |
with
Proof. Condition (23) ensures that each entry of the
Example 1. Consider the model of [20], analysed in the framework of our approach in [2,Example 5.12], i.e.,
\begin{equation} {\partial}_tp^{j}_1 + K^j {\partial}_x p^j_2 = 0, \quad {\partial}_tp^{j}_2 + L^j {\partial}_x p^j_1 = 0, \end{equation} | (24) |
for
\boldsymbol{p_1}( \mathbf{v}) \in X_{ \mathbf{v}}, \quad T_{ \mathbf{v}}\boldsymbol{p_2}( \mathbf{v}) \in X^\perp_{ \mathbf{v}}, |
that is, denoting
\begin{equation} \sum\limits_{j\in J_{ \mathbf{v}}}\phi^j_r p^j_1( \mathbf{v}) = 0, \quad r \in I_2, \quad \sum\limits_{j\in J_{ \mathbf{v}}} \varphi^j_r \nu^j( \mathbf{v})p^j_2( \mathbf{v}) = 0, \qquad r \in I_1, \end{equation} | (25) |
where
\begin{align*} p^r_1(0) & = u^r_1(0)+ u^r_2(0) = 0, \quad r = n_{ \mathbf{v}}+1, \ldots, |J_{ \mathbf{v}}|, \\ p^r_2(0) & = u^r_1(0)- u^r_2(0) = 0, \quad r = 1, \ldots, n_{ \mathbf{v}}. \end{align*} |
Thus
On the other hand, the Kirchhoff condition,
\begin{equation} \sum\limits_{j\in J_{ \mathbf{v}}} \nu^{j}( \mathbf{v}) p^{j}_2( \mathbf{v}) = 0, \end{equation} | (26) |
see [20,Eqn (4)], satisfies the assumption of Proposition 2, as we have
\begin{align*} 0& = \sum\limits_{j\in J_{ \mathbf{v}}}\nu^j( \mathbf{v})p^j_2( \mathbf{v}) = \sum\limits_{j\in J_{ \mathbf{v}}}\nu^j( \mathbf{v}) (f^j_{+, 2}( \mathbf{v})u^j_1( \mathbf{v}) +f^j_{-, 2}( \mathbf{v})u^j_2( \mathbf{v})) \\& = \sum\limits_{j\in J_{ \mathbf{v}}}\nu^j( \mathbf{v})\sqrt{K^jL^j} (u^j_1( \mathbf{v}) -u^j_2( \mathbf{v})) \\ & = -\sum\limits_{j \in J^0_{ \mathbf{v}}} \sqrt{K^jL^j}u^j_1(0) -\sum\limits_{j\in J^1_{ \mathbf{v}}} \sqrt{K^jL^j}u^j_2(1) \\ &\phantom{x}+\sum\limits_{j \in J^1_{ \mathbf{v}}} \sqrt{K^jL^j}u^j_1(1) +\sum\limits_{j\in J^0_{ \mathbf{v}}} \sqrt{K^jL^j}u^j_2(0), \end{align*} |
where we used [2,Eqn 5.2]. Hence, by Proposition 2, Assumption 2 is satisfied.
Example 2. Let us consider the linearized Saint-Venant system,
\begin{equation} {\partial}_tp^j_1 = -V^j {\partial}_x p^j_1 - H^j{\partial}_x p^j_2, \quad {\partial}_t p^j_2 = -g {\partial}_x p^j_1 -V^j {\partial}_x p^j_2, \end{equation} | (27) |
see [2,Example 1.2], assuming that on each edge we have
\begin{equation} \binom{p^j_1}{p^j_2} = \binom{f^j_{+, 1} u^j_1 + f^j_{-, 1}u^j_2}{f^j_{+, 2}u^j_1 + f^j_{-, 2}u^j_2} = \binom{H^j u^j_1+ H^j u^j_2}{\sqrt{gH^j}u^j_1 -\sqrt{gH^j}u^j_2}. \end{equation} | (28) |
We use the flow structure of [11,Example 5.1], shown in Fig. 1, and focus on
p^j_1(0) = p^1_1(1), \quad p^j_2(0) = p_2^1(1), \qquad j = 2, \ldots, N. |
In terms of the Riemann invariants, they can be written as
\begin{align*} &\left(\begin{array}{ccccccc}H^2&H^2&0&0&\ldots&0&0\\\sqrt{gH^2}&-\sqrt{gH^2}&0&0&\ldots&0&0\\\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&0&\ldots&H^N&H^N\\0&0&0&0&\ldots&\sqrt{gH^N}&-\sqrt{gH^N}\end{array}\right)\left(\begin{array}{c}u_1^2(0)\\u_2^2(0)\\\vdots \\ u_1^N(0)\\u_2^N(0)\end{array}\right)\\ &\phantom{xxxx} = \left(\begin{array}{cc}H^1&H^1\\\sqrt{gH^1}&-\sqrt{gH^1}\\\vdots\\ H^1&H^1\\\sqrt{gH^1}&-\sqrt{gH^1}\end{array}\right)\left(\begin{array}{c}u_1^1(1)\\u_2^1(1)\end{array}\right) \end{align*} |
and it is clear that Assumption 2 is satisfied.
For a matrix
\begin{equation} A = (\boldsymbol{a}^c_{j})_{1\leq j\leq m} = (\boldsymbol{a}^r_{i})_{1\leq i\leq n}, \end{equation} | (29) |
that is, we represent the matrix as a row vector of its columns or a column vector of its rows. In particular, we write
\begin{align*} \boldsymbol{\Xi}_{out} & = (\xi^{out}_{ij})_{1\leq i\leq 2m, 1\leq j\leq 2m} = (\boldsymbol{\xi}^{out, c}_{j})_{1\leq j\leq 2m} = (\boldsymbol{\xi}^{out, r}_{i})_{1\leq i\leq 2m}, \\\boldsymbol{\Xi}_{in} & = (\xi^{in}_{ij})_{1\leq i\leq 2m, 1\leq j\leq 2m} = (\boldsymbol{\xi}^{in, c}_{j})_{1\leq j\leq 2m} = (\boldsymbol{\xi}^{in, r}_{i})_{1\leq i\leq 2m}. \end{align*} |
For any vector
Definition 3.5. We say that the problem (1) is graph realizable if there is a graph
Before we formulate the main theorem, we need to introduce some notation. Let us recall that we consider the boundary system (2), i.e.,
\boldsymbol{\Xi}_{out}(( \upsilon_j(0, t))_{j\in J^+}, ( \varpi_j(1, t))_{j\in J^-}) = - \boldsymbol{\Xi}_{in}(( \upsilon_j(1, t))_{j\in J^+}, ( \varpi_j(0, t))_{j\in J^-}). |
Let us emphasize that in this notation, the column indices on the left and right hand side correspond to the values of the same function. To shorten notation, let us renumber them as
In the second step we will determine additional assumptions that allow
Since we do not want (2) to be under- or over-determined, we adopt
Assumption 4. For all
\boldsymbol{\xi}^{out, c}_j\neq 0\quad\mathit{\text{and}}\quad \boldsymbol{\xi}^{out, r}_j \neq 0. |
Our strategy is to treat
Assumption 5. The matrix
\mathsf{A}: = \widehat {\left(\widehat{\boldsymbol{\Xi}_{out}}\right)^T\widehat{\boldsymbol{\Xi}_{in}}} |
is the adjacency matrix of the line graph of a multi digraph.
For
I : = \{i\in \{1, \ldots, 2m\};\; \boldsymbol{\xi}^{in, r}_i = 0\} |
and adopt
Assumption 6. For all
\mathit{\text{supp}}\; \boldsymbol{\xi}^{out, r}_i \subset V^{out}_j. |
In the next proposition we shall show that
Proposition 3. If Assumptions 4, 5 and 6 are satisfied, then the sets
\begin{equation} \{\{\mathcal{V_i}\}_{1\leq i\leq n}, \mathcal{V_S}\}, \end{equation} | (30) |
where
\begin{align} \mathcal{V_S} & = \{i \in \{1, \ldots, 2m\};\; \mathrm{supp\;} \boldsymbol{\xi}^{out, r}_i \subset V^{out}_{M'}\}, \end{align} | (31) |
\begin{align} \mathcal{V_i} & = \bigcup\limits_{s \in V^{out}_i}\mathrm{supp\;} \boldsymbol{{\xi}}^{out, c}_s, \;1\leq i\leq n, \end{align} | (32) |
form a partition of the row indices of both
Since the proof is quite long, we first present its outline.
Step 1. Reconstruct a multi digraph
Step 2. Identify the rows of
Step 3. Associate other rows of
Step 4. Associate remaining rows with vertices and construct a possible partition of the row indices.
Step 5. Check that the constructed partition has the required properties.
Proof. Step 1. By Assumption 5,
Let us recall that the entry
\mathsf{a}_{ij} = \widehat{\widehat{\boldsymbol{{\xi}}^{out, c}_i}\cdot \widehat{\boldsymbol{{\xi}}^{in, c}_j}} |
and if a row
Step 2. To determine the rows in
\mathcal{V_S} = \{i \in \{1, \ldots, 2m\};\; \text{supp}\; \boldsymbol{\xi}^{out, r}_i \subset V^{out}_{M'}\}. |
For any
Step 3. Now, consider the indices
Step 4. Next, we associate the remaining rows in
\mathcal{V}^{out}_i = \bigcup\limits_{s \in V^{out}_i} \text{supp}\; \boldsymbol{{\xi}}^{out, c}_s, \qquad \mathcal{V}^{in}_j = \bigcup\limits_{q \in V^{in}_j} \text{supp}\; \boldsymbol{{\xi}}^{in, c}_q. |
We first observe that if
\begin{equation} \mathcal{V}^{out}_i\setminus \{s \in \mathcal{V}^{out}_i;\; \boldsymbol{\xi}^{in, r}_s = 0\} = \mathcal{V}^{in}_j. \end{equation} | (33) |
Indeed, let
Step 5. We easily check that this partition satisfies the conditions of the proposition. We have already checked this for
We note that (30) does not contain rows corresponding to sinks and they must be added following the rules described in Appendix A. With such an extension, we consider the multi digraph
\begin{equation} \{\{\mathcal{V_i}\}_{1\leq i\leq n}, \mathcal{V_S}, \mathcal{V_Z}\}, \quad \{\{V^{out}_i\}_{1\leq i\leq n}, V^{out}_{M'}, \emptyset\}, \quad \{\{V^{in}_{j_i}\}_{1\leq i\leq n}, \emptyset, V^{in}_{N'}\}, \end{equation} | (34) |
where the association
\begin{equation} \begin{split} &\boldsymbol{\Xi}^i_{out}(( \upsilon_j(0, t))_{j\in J^+\cap V^{out}_i}, ( \varpi_j(1, t))_{j\in J^-\cap V^{out}_i})\\& \phantom{xxx} = - \boldsymbol{\Xi}^i_{in}(( \upsilon_j(1, t))_{j\in J^+\cap V^{in}_{j_i}}, ( \varpi_j(0, t))_{j\in J^-\cap V^{in}_{j_i}}), \quad 1\leq i\leq n, \\ &\boldsymbol{\Xi}^S_{out}(( \upsilon_j(0, t))_{j\in J^+\cap V^{out}_{M'}}, ( \varpi_j(1, t))_{j\in J^-\cap V^{out}_{M'}}) = 0. \end{split} \end{equation} | (35) |
This system can be seen as a Kirchhoff system on the multi digraph
Let
If we grouped all sources into one node, as before Proposition 5, then, by Lemma 3.4, the flow connectivity in this source would be given by
\mathsf{C}_{ \mathbf{v}}: = \widehat{\left(\widehat{\boldsymbol{\Xi}_{out}^{S}}\right)^T\widehat{\boldsymbol{\Xi}_{out}^{S}}}. |
However, such a matrix would not necessarily satisfy Assumption 3. Thus, we separate the arcs into non-communicating groups, each determining a source satisfying Assumption 3. For this, by simultaneous permutations of rows and columns,
\begin{equation} \boldsymbol{\Xi}^S = {\rm diag}\{ \boldsymbol{\Xi}^S_{i}\}_{1\leq i\leq k}, \end{equation} | (36) |
where
\xi^{out, r}_{S_i j} = \left\{\begin{array}{lcl} 1 &\text{if}& j\in V^{out}_{S_i}, \\0&\text{otherwise}.&\end{array}\right. |
For the sinks, it is simpler as there is no constraining information from (2). We have columns with indices in
\begin{equation} V^{in}_{\mathcal{V_i}} = \{j\in V^{in}_{N'};\; \text{supp}\; \boldsymbol{\xi}^{out, c}_j\cap \mathcal{V_i}\neq \emptyset\}, \quad i = 1, \ldots, n, S_1, \ldots, S_k. \end{equation} | (37) |
For each
\begin{equation} V^{in}_{\mathcal{V_i}} = V^{in}_{i, {1}}\cup\ldots\cup V^{in}_{i, {l_i}}, \quad i = 1, \ldots, n, S_1, \ldots, S_k, \end{equation} | (38) |
where
\xi^{in, r}_{\{i, {l}\}, q} = \left\{\begin{array}{lcl} 1 &\text{if}& q\in V^{in}_{i, {l}}, \\0&\text{otherwise}.&\end{array}\right. |
Remark 4. We expect
Then, as in Remark 5, the incoming and outgoing incidence matrices are
A^+ = \left(\begin{array}{c} \mathsf{A}^+\\\boldsymbol{0}\\\boldsymbol{\xi}^{in, r}\end{array}\right), \qquad A^- = \left(\begin{array}{c} \mathsf{A}^-\\\boldsymbol{\xi}^{out, r}\\\boldsymbol{0}\end{array}\right) |
which, by a suitable permutation of columns moving the sources and the sinks to the last positions, can be written as
(39) |
respectively. Both matrices have
(40) |
where the dimensions of the blocks in the first row are, respectively,
Consider a nonzero pair
\begin{equation} \begin{split} &(a_{ij}\mapsto \{k^{ij}_1, \ldots, k^{ij}_h\}, a_{ji}\mapsto \{k^{ji}_1, \ldots, k^{ji}_e\}) \\ & = (a_{ij}\mapsto \text{supp}\; \boldsymbol{a}^{+, r}_i\cap \text{supp}\; \boldsymbol{a}^{-, r}_j, a_{ji}\mapsto \text{supp}\; \boldsymbol{a}^{+, r}_j\cap \text{supp}\; \boldsymbol{a}^{-, r}_i)\end{split} \end{equation} | (41) |
of columns of
Theorem 3.6. System (2) is graph realizable with generalized Kirchhoff's conditions satisfying Assumptions 1 and 2 for
1. for any
\begin{equation} (2, 0), \; (1, 1), \; (0, 2)\; \mathit{\text{or}}\; (0, 0); \end{equation} | (42) |
2. if
\begin{array}{l} if\;({a_{ij}}, {a_{ji}}) = (2, 0)\;or\;(0, 2), \;\;\;then\;k, l \in {J^ + }\;or\;k, l \in {J^ - }\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;and\;{c_k} \ne {c_l}, \\ if\;({a_{ij}}, {a_{ji}}) = (1, 1), \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;then\;k \in {J^ + }\;and\;l \in {J^ - }\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;or\;k \in {J^ - }\;and\;l \in {J^ + }. \end{array} | (43) |
Proof. Necessity. Let us consider the Kirchhoff system (17). By construction, both matrices
\tilde A = (\tilde a_{ij})_{1\leq i, j\leq 2m} = \widehat{(\widehat{\boldsymbol{\Psi}^{out}})^T\widehat{\boldsymbol{\Psi}^{in}}} |
is block diagonal with blocks of the form
\boldsymbol{\Xi}_{out} = \boldsymbol{\Psi}^{out}P, \qquad \boldsymbol{\Xi}_{in} = \boldsymbol{\Psi}^{in}Q, |
where
A = (a_{ij})_{1\leq i, j\leq 2m} : = \widehat{(\widehat{\boldsymbol{\Xi}_{out}})^T\widehat{\boldsymbol{\Xi}_{in}}} = \widehat{(\widehat{\boldsymbol{\Psi}^{out}}P)^T\widehat{\boldsymbol{\Psi}^{in}}Q} = P^T\tilde AQ |
is a matrix where the indices
Sufficiency. Given (2), we have flows
Now, (42) ensures that there are no loops at vertices and that between any two vertices there are either two arcs or none. If
Finally, the assumption
Example 3. Let us consider the system
\begin{equation} \begin{split} {\partial}_t \upsilon_{j}+c_j{\partial}_x \upsilon_j & = 0, \quad 1\leq j\leq 4, \\ {\partial}_t \varpi_{j}-c_j{\partial}_x \varpi_j & = 0, \quad 5\leq j\leq 6, \end{split} \end{equation} | (44) |
where
\begin{equation} \left(\!\!\begin{array}{cccccc}0&1&1&0&0&0\\ 1&0&0&0&0&0\\ 1&1&0&0&0&0\\ 0&0&1&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array}\!\! \right)\left(\!\!\!\begin{array}{c} \upsilon_1(0)\\ \upsilon_2(0)\\ \upsilon_3(0)\\ \upsilon_4(0)\\ \varpi_5(1)\\ \varpi_6(1)\end{array}\!\!\!\right) = \left(\!\!\begin{array}{cccccc}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 1&1&0&0&0&1\\ 0&0&1&1&1&0\end{array}\!\! \right)\left(\!\!\!\begin{array}{c} \upsilon_1(1)\\ \upsilon_2(1)\\ \upsilon_3(1)\\ \upsilon_4(1)\\ \varpi_5(0)\\ \varpi_6(0)\end{array}\!\!\!\right). \end{equation} | (45) |
Thus,
A = \widehat{(\widehat{\boldsymbol{\Xi}_{out}})^T\widehat{\boldsymbol{\Xi}_{in}}} = \left(\begin{array}{cccccc}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 1&1&0&0&0&1\\ 0&0&1&1&1&0\end{array} \right). |
Thus, there is a multi digraph
\boldsymbol{\Xi}^S_{out} = \left(\begin{array}{cccc}0&1&1&0\\ 1&0&0&0\\ 1&1&0&0\\ 0&0&1&1\end{array}\right) \quad\text{and so}\quad \boldsymbol{\Xi}^S = \widehat{\left(\widehat{\boldsymbol{\Xi}_{out}^{S}}\right)^T\widehat{\boldsymbol{\Xi}_{out}^{S}}} = \left(\begin{array}{cccc}1&1&0&0\\ 1&1&1&0\\ 0&1&1&1\\ 0&0&1&1\end{array}\right). |
This matrix is irreducible and thus we have one source. Therefore
A^+ = \left(\begin{array}{cccccc}1&1&0&0&0&1\\ 0&0&1&1&1&0\\ 0&0&0&0&0&0\end{array}\right), \qquad A^- = \left(\begin{array}{cccccc}0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 1&1&1&1&0&0\end{array}\right) |
and consequently
Further,
\begin{align*} \text{supp}\; \boldsymbol{a}^{+, r}_1& = \{1, 2, 6\}, \; \text{supp}\; \boldsymbol{a}^{+, r}_2 = \{3, 4, 5\}, \\ \text{supp}\; \boldsymbol{a}^{-, r}_1& = \{5\}, \; \text{supp}\; \boldsymbol{a}^{-, r}_2 = \{6\}, \; \text{supp}\; \boldsymbol{a}^{-, r}_3 = \{1, 2, 3, 4\}, \end{align*} |
hence, by (41),
(a_{12}\mapsto \{6\}, a_{21}\mapsto \{5\}), \quad (a_{13}\mapsto \{1, 2\}, a_{31}\mapsto \emptyset), \quad (a_{23}\mapsto \{3, 4\}, a_{32}\mapsto \emptyset). |
To reconstruct
Consider a small modification of (44), (45),
\begin{equation} \begin{split} {\partial}_t \upsilon_{j}+c_j{\partial}_x \upsilon_j & = 0, \quad 1\leq j\leq 5, \\ {\partial}_t \varpi_{6}-c_6{\partial}_x \varpi_6 & = 0, \end{split} \end{equation} | (46) |
\begin{equation} \begin{split} \upsilon_5(0)- \upsilon_1(1)- \upsilon_2(1)- \varpi_6(0)& = 0, \\ \varpi_6(1)- \upsilon_3(1)- \upsilon_4(1)- \upsilon_5(1)& = 0. \end{split} \end{equation} | (47) |
The matrices
\begin{equation} \begin{array}{c} {\partial}_t u^1_1+c_1{\partial}_x u^1_1 = 0, \\ {\partial}_t u^1_2+c_2{\partial}_x u^1_2 = 0, \end{array}\;\begin{array}{c} {\partial}_t u^2_1+c_5{\partial}_x u^2_1 = 0, \\ {\partial}_t u^2_2-c_6{\partial}_x u^2_2 = 0, \end{array}\; \begin{array}{c}{\partial}_t u^3_1+c_3{\partial}_x u^3_1 = 0, \\{\partial}_t u^3_2+c_4{\partial}_x u^3_2 = 0, \end{array} \end{equation} | (48) |
with boundary conditions at
(49) |
Consider a digraph
Proposition 4. [6,Thm. 2.4.1] A binary matrix
For our analysis, it is important to understand the reconstruction of
\mathsf{A} = (\mathsf{{a}}_{ij})_{1\leq i, j\leq m} = (\boldsymbol{{a}}^c_j)_{1\leq j\leq m} = (\boldsymbol{{a}}^r_i)_{1\leq i\leq m}. |
If for some
Using the adjacency matrix of a line digraph, we cannot determine how many sources or sinks the original graph could have without additional information. We can lump all potential sources and sinks into one source and one sink, we can have as many sinks and sources as there are zero columns and rows, respectively, or we can subdivide the arcs into some intermediate arrangement. We describe a construction with one source and one sink and indicate its possible variants.
We introduce
\begin{equation} \begin{split} M: = \left\{\begin{array}{lcl} M' &\text{if}& V^{out}_{M'} = \{j;\; \boldsymbol{a}^r_j \neq 0\}, \\ M'-1 &\text{if} & V^{out}_{M'} = \{j;\; \boldsymbol{a}^r_j = 0\}, \end{array}\right.\\ N: = \left\{\begin{array}{lcl} N' &\text{if}& V^{in}_{N'} = \{j;\; \boldsymbol{a}^c_j \neq 0\}, \\ N'-1 &\text{if} & V^{in}_{N'} = \{j;\; \boldsymbol{a}^c_j = 0\}.\end{array}\right.\end{split} \end{equation} | (A.1) |
Thus, we see that the number of internal (or transient) vertices, that is, which are neither sources nor sinks is
\begin{equation} \mathbf{v}_j = \{V^{in}_j, V^{out}_i\}, \qquad a_{i_pj_r} = 1\;\text{for some/any }i_p \in V^{out}_i, j_r \in V^{in}_j. \end{equation} | (A.2) |
With this notation, we present a more algorithmic way of reconstructing
\begin{equation} \mathsf{A}^+ = (\boldsymbol{a}^r_i)_{i\in \mathbb{I}^+}, \qquad \mathsf{A}^- = \left((\boldsymbol{a}^c_j)_{j\in \mathbb{I}^-}\right)^T. \end{equation} | (A.3) |
We see now that each row of
Proposition 5.
Proof. Since each column of
The adjacency matrix
Remark 5. Assume that
where
\bar A^+(\bar A^-)^T = (A^+P)(A^-P)^T = A^+PP^T(A^-)^T = A^+A^-, |
as
(A.4) |
Example 4 Consider the networks
\begin{equation} \mathsf{A} = \left(\begin{array}{ccccccc}0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 1&1&0&1&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&0&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&0&0&0&0\end{array}\right). \end{equation} | (A.5) |
Then,
\begin{equation} \mathsf{A}^+ = \left(\begin{array}{ccccccc} 1&1&0&1&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&0&0&0&0\end{array}\right) \end{equation} | (A.6) |
and we see that there are two transient (internal) vertices
\begin{equation} \mathsf{A}^- = \left(\begin{array}{ccccccc} 0&0&1&0&0&0&0\\ 0&0&0&1&0&1&0\\ 0&0&0&0&0&0&0\end{array}\right). \end{equation} | (A.7) |
The last row corresponds to sinks and the zero columns inform us that arcs
If we want to reconstruct the original graph with one source and one sink, then
A^+ = \left(\begin{array}{ccccccc} 1&1&0&1&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&1&1\end{array}\right), \qquad A^- = \left(\begin{array}{ccccccc} 0&0&1&0&0&0&0\\ 0&0&0&1&0&1&0\\ 1&1&0&0&1&0&1\\ 0&0&0&0&0&0&0\end{array}\right) |
and
A^+(A^-)^T = \left(\begin{array}{cccc} 0&1&2&0\\ 1&0&1&0\\ 0&0&0&0\\ 0&1&1&0\end{array}\right), |
which describes the right multi digraph in Fig. 5. On the other hand, we can consider two sinks (maximum number, as there are two zero columns in
A^+ = \left(\begin{array}{ccccccc} 1&1&0&1&0&0&0\\ 0&0&1&0&1&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1\\ 0&0&0&0&0&1&0\end{array}\right), \qquad A^- = \left(\begin{array}{ccccccc} 0&0&1&0&0&0&0\\ 0&0&0&1&0&1&0\\ 1&0&0&0&0&0&1\\ 0&1&0&0&0&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\end{array}\right) |
and
A^+(A^-)^T = \left(\begin{array}{ccccccc} 0&1&1&1&0&0&0\\ 1&0&0&0&1&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&1&0&0&0&0&0\end{array}\right) |
which describes the left multi digraph in Fig. 5.
It is easily seen that both digraphs have the same line digraph, shown on Fig. 6, whose adjacency matrix is
[1] |
M. F. Nadeem, M. Azeem, A. Khalil, The locating number of hexagonal möbius ladder network, J. Appl. Math. Comput., 66 (2021), 149–165. https://doi.org/10.1007/s12190-020-01430-8 doi: 10.1007/s12190-020-01430-8
![]() |
[2] |
A. Ahmad, A. N. Koam, M. Siddiqui, M. Azeem, Resolvability of the starphene structure and applications in electronics, Ain Shams Eng. J., 13 (2022), 101587. https://doi.org/10.1016/j.asej.2021.09.014 doi: 10.1016/j.asej.2021.09.014
![]() |
[3] |
M. F. Nadeem, M. Imran, H. M. A. Siddiqui, M. Azeem, A. Khalil, Y. Ali, Topological aspects of metal-organic structure with the help of underlying networks, Arab. J. Chem., 14 (2021), 103157. https://doi.org/10.1016/j.arabjc.2021.103157 doi: 10.1016/j.arabjc.2021.103157
![]() |
[4] | M. Azeem, M. F. Nadeem, Metric-based resolvability of polycyclic aromatic hydrocarbons, Eur. Phys. J. Plus, 136 (2021), 395. |
[5] |
M. Imran, A. Ahmad, Y. Ahmad, M. Azeem, Edge weight based entropy measure of different shapes of carbon nanotubes, IEEE Access, 9 (2021), 139712–139724. https://doi.org/10.1109/ACCESS.2021.3119032 doi: 10.1109/ACCESS.2021.3119032
![]() |
[6] |
M. F. Nadeem, A. Shabbir, Computing and comparative analysis of topological invariants of y-junction carbon nanotubes, Int. J. Quant. Chem., 122 (2022), e26847. https://doi.org/10.1002/qua.26847 doi: 10.1002/qua.26847
![]() |
[7] |
X. Zuo, M. F. Nadeem, M. K. Siddiqui, M. Azeem, Edge weight based entropy of different topologies of carbon nanotubes, IEEE Access, 9 (2021), 102019–102029. https://doi.org/10.1109/ACCESS.2021.3097905 doi: 10.1109/ACCESS.2021.3097905
![]() |
[8] |
M. F. Nadeem, M. Azeem, H. M. K. Siddiqui, Comparative study of zagreb indices for capped, semi-capped and uncapped carbon nanotubes, Polycyclic Aromat. Compd., 42 (2022), 3545–3562. https://doi.org/10.1080/10406638.2021.1890625 doi: 10.1080/10406638.2021.1890625
![]() |
[9] |
F. Afzal, S. Hussain, D. Afzal, S. Razaq, Some new degree based topological indices via m-polynomial, J. Inf. Optim. Sci., 41 (2020), 1061–1076. https://doi.org/10.1080/02522667.2020.1744307 doi: 10.1080/02522667.2020.1744307
![]() |
[10] | A. Rauf, M. Naeem, S. U. Bukhari, Quantitative structure–property relationship of ev-degree and ve-degree based topological indices with physico-chemical properties of benzene derivatives and application, Int. J. Quant. Chem., 122 (2022), e26851. |
[11] | A. Rauf, M. Naeem, A. Aslam, Quantitative structure–property relationship of edge weighted and degree-based entropy of benzene derivatives, Int. J. Quant. Chem., 122 (2022), e26839. |
[12] | J. B. Liu, X.-B. Peng, S. Hayat, Topological index analysis of a class of networks analogous to alicyclic hydrocarbons and their derivatives, Int. J. Quant. Chem., 122 (2022), e26827. |
[13] |
Y. Shang, Sombor index and degree-related properties of simplicial networks, Appl. Math. Comput., 419 (2022), 126881. https://doi.org/10.1016/j.amc.2021.126881 doi: 10.1016/j.amc.2021.126881
![]() |
[14] |
Z. Wang, Y. Mao, K. C. Das, Y. Shang, Nordhaus–gaddum-type results for the steiner gutman index of graphs, Symmetry, 12 (2020), 1711. https://doi.org/10.3390/sym12101711 doi: 10.3390/sym12101711
![]() |
[15] |
Y. Shang, Lower bounds for gaussian estrada index of graphs, Symmetry, 10 (2018), 325. https://doi.org/10.3390/sym10080325 doi: 10.3390/sym10080325
![]() |
[16] |
S. Khan, S. Pirzada, Y. Shang, On the sum and spread of reciprocal distance laplacian eigenvalues of graphs in terms of harary index, Symmetry, 14 (2022), 1937. https://doi.org/10.3390/sym14091937 doi: 10.3390/sym14091937
![]() |
[17] | J. B. Liu, J. J. Gu, K. Wang, The expected values for the gutman index, schultz index, and some sombor indices of a random cyclooctane chain, Int. J. Quant. Chem., 123 (2022). |
[18] |
M. Azeem, M. Imran, M. F. Nadeem, Sharp bounds on partition dimension of hexagonal möbius ladder, J. King Saud Univ. Sci., 34 (2022), 101779. https://doi.org/10.1016/j.jksus.2021.101779 doi: 10.1016/j.jksus.2021.101779
![]() |
[19] |
A. Shabbir, M. Azeem, On the partition dimension of tri-hexagonal alpha-boron nanotube, IEEE Access, 9 (2021), 55644–55653. https://doi.org/10.1109/ACCESS.2021.3071716 doi: 10.1109/ACCESS.2021.3071716
![]() |
[20] |
J. B. Liu, M. F. Nadeem, M. Azeem, Bounds on the partition dimension of convex polytopes, Comb. Chem. High Throughput Screening, 25 (2022), 547–553. https://doi.org/10.2174/1386207323666201204144422 doi: 10.2174/1386207323666201204144422
![]() |
[21] | J. B. Liu, Y. Bao, W. T. Zheng, S. Hayat, Network coherence analysis on a family of nested weighted n-polygon networks, Fractals, 29 (2021), 2150260. |
[22] | J. B. Liu, J. J. Gu, Computing and analyzing the normalized laplacian spectrum and spanning tree of the strong prism of the dicyclobutadieno derivative of linear phenylenes, Int. J. Quant. Chem., 122 (2022), e26972. |
[23] |
J. B. Liu, J. Zhao, Z. Q. Cai, On the generalized adjacency, laplacian and signless laplacian spectra of the weighted edge corona networks, Phys. A Stat. Mechan. Appl., 540 (2020), 123073. https://doi.org/10.1016/j.physa.2019.123073 doi: 10.1016/j.physa.2019.123073
![]() |
[24] | J.-B. Liu, J. Zhao, J. Min, J. Cao, The hosoya index of graphs formed by a fractal graph, Fractals, 27 (2019), 1950135. |
[25] |
J.-B. Liu, C. Wang, S. Wang, and B. Wei, Zagreb indices and multiplicative zagreb indices of eulerian graphs, Bull. Malays. Math. Sci. Soc., 42 (2017), 67–78. https://doi.org/10.1007/s40840-017-0463-2 doi: 10.1007/s40840-017-0463-2
![]() |
[26] | J. B. Liu, X. F. Pan, Minimizing kirchhoff index among graphs with a given vertex bipartiteness, Appl. Math. Comput., 291 (2016), 84–88. |
[27] |
J. B. Liu, X. F. Pan, F. T. Hu, F. F. Hu, Asymptotic laplacian-energy-like invariant of lattices, Appl. Math. Comput., 253 (2015), 205–214. https://doi.org/10.1016/j.amc.2014.12.035 doi: 10.1016/j.amc.2014.12.035
![]() |
[28] |
S. Bukhari, M. K. Jamil, M. Azeem, S. Swaray, Patched network and its vertex-edge metric-based dimension, IEEE Access, 11 (2023), 4478–4485. https://doi.org/10.1109/ACCESS.2023.3235398 doi: 10.1109/ACCESS.2023.3235398
![]() |
[29] | M. C. Shanmukha, S. Lee, A. Usha, K. C. Shilpa, M. Azeem, Degree-based entropy descriptors of graphenylene using topological indices, Comput. Model. Eng. Sci., 2023 (2023), 1–25. |
[30] |
X. Zhang, M. T. A. Kanwal, M. Azeem, M. K. Jamil, M. Mukhtar, Finite vertex-based resolvability of supramolecular chain in dialkyltin, Main Group Metal Chem., 45 (2022), 255–264. https://doi.org/10.1515/mgmc-2022-0027 doi: 10.1515/mgmc-2022-0027
![]() |
[31] |
Q. Huang, A. Khalil, D. A. Ali, A. Ahmad, R. Luo, M. Azeem, Breast cancer chemical structures and their partition resolvability, Math. Biosci. Eng., 20 (2022), 3838–3853. https://doi.org/10.3934/mbe.2023180 doi: 10.3934/mbe.2023180
![]() |
[32] |
M. Azeem, M. K. Jamil, A. Javed, A. Ahmad, Verification of some topological indices of y-junction based nanostructures by m-polynomials, J. Math., 2022 (2022), 1–18. https://doi.org/10.1155/2022/8238651 doi: 10.1155/2022/8238651
![]() |
[33] | M. K. Jamil, M. Imran, K. A. Sattar, Novel face index for benzenoid hydrocarbons, Mathematics, 8 (2020), 312. |
[34] |
X. Zhang, A. Raza, A. Fahad, M. K. Jamil, M. A. Chaudhry, Z. Iqbal, On face index of silicon carbides, Discrete Dyn. Nat. Soc., 2020 (2020), 1–8. https://doi.org/10.1155/2020/6048438 doi: 10.1155/2020/6048438
![]() |
[35] |
A. Ye, A. Javed, M. K. Jamil, K. A. Sattar, A. Aslam, Z. Iqbal, et al., On computation of face index of certain nanotubes, Discrete Dyn. Nat. Soc., 2020 (2020), 1–6. https://doi.org/10.1155/2020/3468426 doi: 10.1155/2020/3468426
![]() |
[36] |
Z. Ahmad, , M. Naseem, M. K. Jamil, M. K. Siddiqui, M. F. Nadeem, New results on eccentric connectivity indices of v-phenylenic nanotube, Eurasian Chem. Commun., 2 (2020), 663–671. https://doi.org/10.33945/SAMI/ECC.2020.6.3 doi: 10.33945/SAMI/ECC.2020.6.3
![]() |
[37] |
Z. Ahmad, , M. Naseem, M. K. Jamil, M. F. Nadeem, S. Wang, Eccentric connectivity indices of titania nanotubes TiO, Eurasian Chem. Commun., 2 (2020), 712–721. https://doi.org/10.33945/SAMI/ECC.2020.6.8 doi: 10.33945/SAMI/ECC.2020.6.8
![]() |
[38] |
A. N. A. Koam, A. Ahmad, M. Nadeem, Comparative study of valency-based topological descriptor for hexagon star network, Comput. Syst. Sci. Eng., 36 (2021), 293–306. https://doi.org/10.32604/csse.2021.014896 doi: 10.32604/csse.2021.014896
![]() |
[39] | H. M. A. Siddiqui, S. Baby, M. F. Nadeem, M. K. Shafiq, Bounds of some degree based indices of lexicographic product of some connected graphs, Polycyclic Aromat. Compd., 42 (2022), 2568–2580. |
[40] |
J. B. Liu, H. M. A. Siddiqui, M. F. Nadeem, M. A. Binyamin, Some topological properties of uniform subdivision of sierpiński graphs, Main Group Metal Chem., 44 (2021), 218–227. https://doi.org/10.1515/mgmc-2021-0006 doi: 10.1515/mgmc-2021-0006
![]() |
[41] |
M. Ishtiaq, A. Rauf, Q. Rubbab, M. K. Siddiqui, H. Ibrahim, Algebraic polynomial based topological properties of anti-tumor drug hyaluronic acid-doxorubicin (HAD), Polycyclic Aromat. Compd., 42 (2022), 7049–7070. https://doi.org/10.1080/10406638.2021.1995011 doi: 10.1080/10406638.2021.1995011
![]() |
[42] |
V. Ravi, M. K. Siddiqui, N. Chidambaram, K. Desikan, On topological descriptors and curvilinear regression analysis of antiviral drugs used in COVID-19 treatment, Polycyclic Aromat. Compd., 42 (2022), 6932–6945. https://doi.org/10.1080/10406638.2021.1993941 doi: 10.1080/10406638.2021.1993941
![]() |
[43] | A. Ahmad, S. C. López, Distance-based topological polynomials associated with zero-divisor graphs, Math. Prob. Eng., 2021 (2021), 1–8. |
[44] | A. Ahmad, Vertex-degree based eccentric topological descriptors of zero divisor graph of commutative rings, Online J. Anal. Comb., 15 (2020), 1–10. |
[45] |
A. Ahmad, R. Hasni, K. Elahi, M. A. Asim, Polynomials of degree-based indices for swapped networks modeled by optical transpose interconnection system, IEEE Access, 8 (2020), 214293–214299. https://doi.org/10.1109/ACCESS.2020.3039298 doi: 10.1109/ACCESS.2020.3039298
![]() |
[46] |
F. A. Abolaban, A. Ahmad, M. A. Asim, Computation of vertex-edge degree based topological descriptors for metal trihalides network, IEEE Access, 9 (2021), 65330–65339. https://doi.org/10.1109/ACCESS.2021.3076036 doi: 10.1109/ACCESS.2021.3076036
![]() |
[47] |
Özge Çolakoğlu Havare, Topological indices and QSPR modeling of some novel drugs used in the cancer treatment, Int. J. Quant. Chem., 121 (2021), e26813. https://doi.org/10.1002/qua.26813 doi: 10.1002/qua.26813
![]() |
[48] | J. B. Liu, R. M. Singaraj, Topological analysis of para-line graph of remdesivir used in the prevention of corona virus, Int. J. Quant. Chem., 121 (2021), e26778. |
[49] | M. M. Zobair, M. A. Malik, H. Shaker, Eccentricity-based topological invariants of tightest nonadjacently configured stable pentagonal structure of carbon nanocones, Int. J. Quant. Chem., 121 (2021), e26807. |
[50] |
Z. Sabir, M. Umar, M. A. Z. Raja, H. M. Baskonus, W. Gao, Designing of morlet wavelet as a neural network for a novel prevention category in the HIV system, Int. J. Biomath., 15 (2021), 2250012. https://doi.org/10.1142/S1793524522500127 doi: 10.1142/S1793524522500127
![]() |
[51] |
M. Cancan, D. Afzal, S. Hussain, A. Maqbool, F. Afzal, Some new topological indices of silicate network via m-polynomial, J. Discrete Math. Sci. Cryptography, 23 (2020), 1157–1171. https://doi.org/10.1080/09720529.2020.1809776 doi: 10.1080/09720529.2020.1809776
![]() |
[52] |
J. B. Liu, M. K. Shafiq, H. Ali, A. Naseem, N. Maryam, S. S. Asghar, Topological indices of mth chain silicate graphs, Mathematics, 7 (2019), 42. https://doi.org/10.3390/math7010042 doi: 10.3390/math7010042
![]() |
[53] |
A. Q. Baig, M. Imran, H. Ali, On topological indices of poly oxide, poly silicate, DOX, and DSL networks, Can. J. Chem., 93 (2015), 730–739. https://doi.org/10.1139/cjc-2014-0490 doi: 10.1139/cjc-2014-0490
![]() |
[54] |
M. S. Chen, K. Shin, D. Kandlur, Addressing, routing, and broadcasting in hexagonal mesh multiprocessors, IEEE Trans. Comput., 39 (1990), 10–18. https://doi.org/10.1109/12.46277 doi: 10.1109/12.46277
![]() |
[55] |
M. F. Nadeem, M. Azeem, I. Farman, Comparative study of topological indices for capped and uncapped carbon nanotubes, Polycyclic Aromat. Compd., 42 (2022), 4666–4683. https://doi.org/10.1080/10406638.2021.1903952 doi: 10.1080/10406638.2021.1903952
![]() |
[56] |
M. F. Nadeem, M. Azeem, H. M. A. Siddiqui, Comparative study of zagreb indices for capped, semi-capped, and uncapped carbon nanotubes, Polycyclic Aromat. Compd., 42 (2022), 3545–3562. https://doi.org/10.1080/10406638.2021.1890625 doi: 10.1080/10406638.2021.1890625
![]() |
[57] | M. V. Diudea, C. L. Nagy, Diamond and Related Nanostructures, Springer Netherlands, 2013. https://doi.org/10.1007/978-94-007-6371-5 |
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