A novel edge-weighted matrix of a graph and its spectral properties with potential applications

1.   Introduction

Every graph taken into consideration here is a simple graph $\Gamma = (V, E)$, where $|V| = \nu$ and $|E| = m$. The count of edges connected to a vertex $v \in V$, denoted by $d_{ \Gamma}(v)$ is the degree of that vertex. The degree of an edge $e = uv$ is defined by $d_{ \Gamma}(e) = d_{ \Gamma}(u)+d_{ \Gamma}(v)-2$. The parameters $\Delta$ (resp. $\delta$) and $\Delta^{\prime}$($\delta^{\prime}$) represent the maximum (resp. minimum) vertex and edge degree of a graph, respectively. If $\Delta_{ \Gamma} = \delta_{ \Gamma} = \nu-1$ then the graph $\Gamma$ is said to be a complete graph denoted by $K_{\nu}$. The graph $\Gamma$ is called bipartite if $V(\Gamma)$ is partitioned into two sets say, $M$ and $N$ (partite sets), such that every edge in $\Gamma$ has one endpoint in $M$ and the other in $N$. If every vertex of $M$ is adjacent to every vertex of $N$ with $|M| = r$ and $|N| = s$, then the graph is called the complete bipartite graph, denoted as $K_{r, s}$. The graph $K_{1, \nu-1}$ denoted by $S_{\nu}$ is a star graph, and the graph $K_{r, r}$ is called the equi-bipartite graph. The graph $\overline{ \Gamma}$ represents the complement of a graph $\Gamma$, which is defined on the same vertex set as $\Gamma$ such that if two vertices are adjacent in $\overline{ \Gamma}$, then they are not adjacent in $\Gamma$. For more graph-theoretic terminologies, we refer to the book by Harary ^[19]. For any real number $x$, the floor function is the greatest integer less than or equal to $x$, denoted as $\lfloor {x}\rfloor$.

The first degree-based molecular descriptor, the Zagreb index, was developed by Gutman and Trinajsti$ć$ ^[9]. It first emerged in the topological formula for conjugated molecules regarding their total $\pi$-electron energy. The first Zagreb index is defined as:

The Zagreb indices were reformulated in 2004 by Mili$\acute{c}$evi$\acute{c}$ et al. ^[28] in terms of edge degree, where the edge degree is given by $d_{ \Gamma}(e) = d_{ \Gamma}(u)+d_{ \Gamma}(v)-2$ for $e = uv \in E(\Gamma)$. Thus, the first reformulated Zagreb index is given by

The reader is referred to ^[9] regarding applications of Zagreb indices.

The sum of the absolute values of eigenvalues of the adjacency matrix of $\Gamma$ gives us the energy $E(\Gamma)$ of a graph $\Gamma$. This quantity is introduced in ^[10]. Suppose $\lambda_{1}\geq \lambda_{2}\geq\ldots\geq \lambda_{\nu}$ are the eigenvalues of the adjacency matrix $A(\Gamma)$, then the energy of the graph $\Gamma$ is given by

Other graph energies were also introduced and studied. Indulal et al. ^[20] presented some results on the distance energy of graphs. Gutman and Zhou ^[8] investigated the Laplcaian energy of graphs and derived some extremal results from it.

The extended adjacency matrix of graph $\Gamma$ was proposed by Yang et al. ^[36] in 1994, denoted by $A_{ex}(\Gamma)$. Its $(i, j)$-entry is defined to be equal to $\frac{1}{2}\left(\frac{d_{j}}{d_{i}}+\frac{d_{i}}{d_{j}}\right)$ if $v_{i}\sim v_{j}$ and 0 otherwise. Since $A_{ex}$ is a real symmetric matrix of order $\nu$, all its eigenvalues are real, which are denoted as $\eta_{1}\geq \eta_{2}\geq \ldots\geq \eta_{\nu}$. Yang et al. ^[36] also investigated the extended graph energy by summing the absolute values of the eigenvalues of the $A_{ex}$-matrix, defined as

For recent studies on graph energy, we refer to ^{[5,6,23,24,33,35]}.

1.1.   Edge-weighted graph energy

For a given graph $\Gamma$, we define the following terminologies:

Definition 1.1. Let $V = \{v_{1}, v_{2}, \ldots, v_{\nu}\}$ be the vertex set and $\{w_{1}, w_{2}, \ldots, w_{\nu}\}$ be the vertex-weights, then the vertex weight for $v_{i}\in V$ is defined as $w(v_{i}) = d_{ \Gamma}(v_{i})$. The range for a vertex-weight of $v_{i}\in V$ is $0\leq w(v_{i})\leq \Delta_{ \Gamma}$ (here $w(v_{i}) = 0$ if $\Gamma$ is disconnected).

Definition 1.2. Let $E_ \Gamma = \{e_{1}, e_{2}, \ldots, e_{m}\}$ be the edge set, then the edge-weight of $e_i = u_iv_i$ is defined by $w(e_i) = d_{ \Gamma}(u_i)+d_{ \Gamma}(v_i)-2$. The range for an edge-weight of $e\in E$ is $0\leq w(e)\leq \frac{M_{1}(\Gamma)}{2}-m$ (where $w(e) = 0$ if and only if $\Gamma = K_2$.

Definition 1.3. The weighted degree of a vertex $v \in V(\Gamma)$ is defined as

It is very clear that $w(e) = d_{ \Gamma}(e)$ for every $e \in E$.

Observe that

Motivated by the extended adjacency matrix of graph $\Gamma$, we introduce a new edge-weighted adjacency matrix of graph $\Gamma$ denoted by $A_{w}(\Gamma)$. It is defined in such a way that, for any $v_{i}$ that is adjacent to $v_{j}$, its $(i, j)$-entry equals $d_{ \Gamma}(v_{i})+d_{ \Gamma}(v_{j})-2$; otherwise, it equals to 0. In fact, $A_{w}(\Gamma)$ is a real symmetric matrix of order $\nu$. Hence, all its eigenvalues are real and can be arranged as $\lambda_{1}^{w}\geq \lambda_{2}^{w}\geq \ldots \geq \lambda_{\nu}^{w}$, where the largest eigenvalue $\lambda_{1}^{w}$ is called the spectral radius of $A_{w}(\Gamma)$. The edge-weighted graph energy of $\Gamma$ is given by

Following the types of adjacencies in ^[34], the type of adjacency employed by the edge-weighted matrix is the vertex-based adjacency.

1.2.   Some useful identities

This subsection presents some basic properties of edge-weighted eigenvalues of graphs.

$(1)$ $\sum_{i = 1}^{\nu} \lambda_{i}^{w} = 0$.

$(2)$ $\sum_{i = 1}^{\nu} \left(\lambda_{i}^{w}\right)^{2} = 2\sum_{i = 1}^{m}[w(e_{i})]^{2} = 2\sum_{i = 1}^{m}d_{ \Gamma}(e_{i})^{2} = 2EM_{1}(\Gamma)$.

$(3)$ $\sum_{0\leq i\leq j}\lambda_{i}^{w}\lambda_{j}^{w} = -\sum_{i = 1}^{m}[w(e_{i})]^{2} = -\sum_{i = 1}^{m}d_{ \Gamma}(e_{i})^{2} = -EM_{1}(\Gamma)$.

Moreover, observe that,

$(1)$ $\sum_{i = 1}^{\nu}. \left(\lambda_{i}^{w}\right)^{2} = 2EM_{1}(\Gamma)$.

$(2)$ $\sum_{0\leq i\leq j}\lambda_{i}^{w}\lambda_{j}^{w} = -EM_{1}(\Gamma)$.

The following example delivers the edge-weighted energy of standard graph families.

Example 1.4. (1) The edge-weighted energy $E_w(K_\nu)$ of $K_{\nu}$ is $4 (\nu-1)(\nu-2)$.

(2) The edge-weighted energy $E_w(K_{r, s})$ of $K_{r, s}$ is $2 (r+s-2)\sqrt{rs}$.

(3) The edge-weighted energy $E_w(S_\nu)$ of $S_{\nu}$ is $2 (\nu-2)\sqrt{(\nu-1)}$.

(4) The edge-weighted energy $E_w(K_{r, r})$ of $K_{r, r}$ is $4 r(r-1)$.

Proof. The edge-weighted energy $E_w(K_{r, s})$ of $K_{r, s}$ is $2 (r+s-2)\sqrt{rs}$. Replacing $s$ by $r$ in $E_w(K_{r, s})$ of $K_{r, s}$, we get $E_w(K_{r, r})$ of $K_{r, r}$ as $2 (r+r-2)\sqrt{r^{2}} = 4 r(r-1)$. □

2.   Auxiliary results

In subsequent sections, we need the following already established results:

Lemma 2.1. ^[35] Let $C = (c_{ij})$ and $D = (d_{ij})$ be real symmetric non-negative matrices of order $\nu$. If $C\geq D$, i.e., $c_{ij}\geq d_{ij}$ for all $i, j$, then $\lambda_{1}(C)\geq \lambda_{1}(D)$, whereas $\lambda_{1}$ is the largest eigenvalue.

Lemma 2.2. ^[35] Let $\Gamma$ be a connected graph of order $\nu$ with $m$ edges. Then

with equality if and only if $\Gamma\cong K_{1, \nu-1}$ or $\Gamma\cong K_{\nu}$.

Here we deliver the well-known Cauchy–Schwartz inequality.

Lemma 2.3. (Cauchy–Schwartz inequality)^[3] Let $r_{i}$ and $s_{i}$, $1\leq i\leq \nu$ be any real numbers, then

The Ozeki inequality is frequently used in the spectral analysis of graphs.

Lemma 2.4. (Ozeki inequality)^[30] If $r_{i}$ and $s_{i}$ $(1\leq i\leq \nu)$ are non-negative real numbers, then

where $P_{1} = \max_{1\leq i\leq \nu}\{r_{i}\}$, $P_{2} = \max_{1\leq i\leq \nu}\{s_{i}\}$, $p_{1} = \min_{1\leq i\leq \nu}\{r_{i}\}$, $p_{2} = \min_{1\leq i\leq \nu}\{s_{i}\}$.

The following inequality has been retrieved from Dragomir ^[7].

Lemma 2.5. ^[7] Let $p_{i}$, $q_{i}$, $r_{i}$ and $s_{i}$ be sequences of real numbers, and $m_{i}$, $\nu_{i}$ are non-negative for $i = 1, 2, \ldots, \nu$. Then the following inequality is valid:

Jog and Gurjar ^[23] used the following inequality while studying bounds on the distance energy of graphs:

Lemma 2.6. Let $r_{i}$, $1\leq i\leq \nu$ be any real numbers, then

The following inequality was shown by Pólya and Szegó in their book ^[31].

Lemma 2.7. ^[31] Suppose $r_{i}$ and $s_{i}$, $1\leq i\leq \nu$ are positive real numbers, then

where $P_{1} = max_{1\leq i\leq \nu}\{r_{i}\}$, $P_{2} = max_{1\leq i\leq \nu}\{s_{i}\}$, $p_{1} = min_{1\leq i\leq \nu}\{r_{i}\}$, $p_{2} = min_{1\leq i\leq \nu}\{s_{i}\}$.

The following classical inequality was proven by Biernacki et al. ^[2].

Lemma 2.8. ^[2] Suppose $r_{i}$ and $s_{i}$, $1\leq i\leq \nu$ are positive real numbers, then

where $r$, $s$, $R$, and $S$ are real constants such that for each $i$, $1\leq i\leq \nu$, $r\leq r_{i}\leq R$, and $s\leq s_{i}\leq S$. Further, $\alpha(\nu) = \nu\lfloor \frac{\nu}{2}\rfloor \left(1-\frac{1}{\nu}\lfloor \frac{\nu}{2}\rfloor \right)$.

Diaz and Metcalf ^[4] delivered a proof of the following inequality:

Lemma 2.9. ^[4] Let $r_{i}$ and $s_{i}$, $1\leq i\leq \nu$ are nonnegative real numbers, then

where $p$ and $P$ are real constants, so that for each $i$, $1\leq i\leq \nu$, holds, $pr_{i}\leq s_{i}\leq Pr_{i}$.

Next, we prove the following inequality on the edge-connected eigenvalues:

Lemma 2.10. Let $\Gamma$ be a connected graph of order $\nu \geq 2$. Then $\lambda_{1}^{w} > \lambda_{2}^{w}$.

Proof. Let us assume, for the sake of contradiction, that $\lambda_{1}^{w} = \lambda_{2}^{w}$. Since $\Gamma$ is connected, $A_{w}(\Gamma)$ is an irreducible non-negative $\nu \times \nu$ matrix. By the Perron–Frobenius theorem, the eigenvector $x$ corresponding to $\lambda_{1}^{w}$ has all components positive. Let $y$ be an eigenvector corresponding to $\lambda_{2}^{w}$. Since $\lambda_{1}^{w} = \lambda_{2}^{w}$, any linear combination of $x$ and $y$ would be an eigenvector corresponding to $\lambda_{1}^{w}$. This implies that it would be possible to construct an eigenvector with some zero components, which contradicts the fact that all components of $x$ are positive. Hence, we must have $\lambda_{1}^{w} > \lambda_{2}^{w}$. □

Next, we deliver a characterization for an $\nu$-vertex satisfying $|\lambda_{1}^{w}| = |\lambda_{2}^{w}| = \ldots = |\lambda_{\nu}^{w}|$.

Proposition 2.11. Let $\Gamma$ be a graph of order $\nu$. Then $|\lambda_{1}^{w}| = |\lambda_{2}^{w}| = \ldots = |\lambda_{\nu}^{w}|$ if and only if $\Gamma \cong \overline{K_{\nu}}$ or $\Gamma \cong \frac{\nu}{2} K_{2}$.

Proof. First, assume that $|\lambda_{1}^{w}| = |\lambda_{2}^{w}| = \ldots = |\lambda_{\nu}^{w}|$. Let $S$ be the number of isolated vertices in $\Gamma$. If $S \geq 1$, then $\lambda_{1}^{w} = \lambda_{2}^{w} = \ldots = \lambda_{\nu}^{w} = 0$, hence $\Gamma \cong \overline{K_{\nu}}$ or $\Gamma \cong \frac{\nu}{2} K_{2}$. Otherwise, if the maximum degree $\Delta \geq 2$, then $\Gamma$ contains a connected component $H$ with at least 3 vertices. If $H = K_{\nu}$, $\nu \geq 3$, then by Lemma 2.10, $|\lambda_{1}^{w}| = 2(\nu-1)(\nu-2)$ and $|\lambda_{2}^{w}| = 2(\nu-2)$, clearly $|\lambda_{1}^{w}| > |\lambda_{2}^{w}|$, a contradiction. Otherwise, if $H$ is not a complete graph, then by Lemma 2.10, $\lambda_{1}^{w} > \lambda_{2}^{w}$, a contradiction.

Conversely, one can easily check that $|\lambda_{1}^{w}| = |\lambda_{2}^{w}| = \ldots = |\lambda_{\nu}^{w}|$ holds for $\overline{K_{\nu}}$ and $\frac{\nu}{2} K_{2}$. □

3.   Main results

This section delivers various upper/lower extremal values for $E_{w}(\Gamma)$ and $\lambda_{1}^{w}$ for an ($m, \nu$)-graph with $\nu$-vertices and $m$-edges.

A sharp upper bound on $\lambda_{1}^{w}$ (i.e., edge-weighted spectral radius) is being proven in the following result.

Theorem 3.1. Let $\Gamma$ be an ($m, \nu$)-graph possessing the maximum degree $\Delta$. Then

with $\lambda_{1}^{w} = 2\left(\Delta -1\right)\sqrt{1-\nu+2m}\iff$ $\Gamma\cong K_{\nu}$, $\nu\geq 2$.

Proof. Since $A_{w}(\Gamma) \leq 2(\Delta -1)A(\Gamma)$ and $\lambda_{1}^{w}$ be its spectral radius. Employing Lemma 2.1, one has

By Lemma 2.2, one obtains

Also, $\lambda_{1}^{w} = 2\left(\Delta -1\right)\sqrt{1-\nu+2m}$ in (3.1) $\iff \Gamma\cong K_{\nu}$, $\nu\geq 2$. □

The next theorem delivers an upper extremal value considering the first–formulated Zagreb $EM_1$ index of $\Gamma$.

Theorem 3.2. If $\Gamma$ is an $\nu$-vertex graph having $\lambda_{1}^{w}$ as its spectral radius (the largest eigenvalue), then

Proof. Since $\sum_{k = 1}^{\nu}\lambda_{k}^{w} = 0$ it can be rewritten as $\sum_{k = 2}^{\nu}\lambda_{k}^{w} = -\lambda_{1}^{w}$. Further, $\left(\sum_{k = 1}^{\nu}(\lambda_{k}^{w})^{2}\right) = 2EM_{1}(\Gamma)$, $(\sum_{k = 2}^{\nu}(\lambda_{k}^{w})^{2}) = (\sum_{k = 1}^{\nu}(\lambda_{k}^{w})^{2}-(\lambda_{1}^{w})^{2}) = (2EM_{1}(\Gamma)-(\lambda_{1}^{w})^{2})$ and $(\sum_{k = 2}^{\nu}1) = (\nu-1)$.

Put $r_{k} = 1$ and $s_{k} = \lambda_{k}^{w}$ in Lemma 2.3 and we obtain

Hence, we have furnished the proof. □

Some lower and upper extremal values on $E_{w}$ i.e., edge-weighted energy of $\Gamma$.

Theorem 3.3. For a connected $\nu$-vertex graph $\Gamma$, $E_{w}$ satisfies

Proof. For the upper bound, consider Lemma 2.3, i.e.,

put $r_{k} = 1$ and $s_{k} = |\lambda_{k}^{w}|^{2}$ in Lemma 2.3, we obtain

since $\left(\sum_{k = 1}^{\nu}|\lambda_{k}^{w}|\right) = E_{w}(\Gamma)$, $\left(\sum_{k = 1}^{\nu}1^{2}\right) = 1$ and $\left(\sum_{k = 1}^{\nu}(|\lambda_{k}^{w}|)^{2}\right) = 2EM_{1}(\Gamma)$, we have

Similarly, for a lower bound, consider Lemma 2.6, i.e.,

put $r_{k} = |\lambda_{k}^{w}|$ in Lemma 2.6, we obtan

Hence, by combining the upper bound and the lower bound, we obtain the required result. □

In terms of the $EM_{1}$ index, our next result delivers another lower extremal value on $E_{w}$.

Theorem 3.4. Any $(m, \nu)$-graph $\Gamma$ satisfies

where $|\lambda_{\nu}^{w}|$ (resp. $|\lambda_{1}^{w}|$) are minimum (resp. maximum) values of $|\lambda_{k}^{w}|$.

Proof. Let $|\lambda_{1}^{w}|\geq |\lambda_{2}^{w}|\geq \ldots \geq |\lambda_{\nu}^{w}|$ be the eigenvalues of $A_{w}(\Gamma)$. By putting $r_{k} = 1$, $s_{k} = |\lambda_{k}^{w}|$ $P_{1} = 1$, $P_{2} = |\lambda_{1}^{w}|$, $p_{1} = 1$ and $p_{2} = |\lambda_{\nu}^{w}|$ in Lemma 2.4, one gets

since $\left(\sum_{k = 1}^{\nu}(\lambda_{k}^{w})^{2}\right) = 2EM_{1}(\Gamma)$ we have

Hence, the proof has been furnished. □

The following theorem further refines the bound in Theorem 3.4.

Theorem 3.5. For an $(\nu, m)$-graph $\Gamma$, assume $\lambda_{1}^{w}\geq \lambda_{2}^{w}\geq \ldots \geq \lambda_{\nu}^{w}$ are eigenvalues of $A_{w}(\Gamma)$. This implies that,

where $\alpha(\nu) = \nu\lfloor \frac{\nu}{2}\rfloor \left(1-\frac{1}{\nu}\lfloor \frac{\nu}{2}\rfloor \right)$.

Proof. Let $|\lambda_{1}^{w}|\geq |\lambda_{2}^{w}|\geq \ldots \geq |\lambda_{\nu}^{w}|$ be the eigenvalues of $A_{w}(\Gamma)$. By putting $r_{k} = |\lambda_{k}^{w}| = s_{k}$, $R = |\lambda_{1}^{w}| = S$, and $r = |\lambda_{\nu}^{w}| = s$ in Lemma 2.8, one obtains

Hence the proof has been furnished. □

For a non-zero eigenvalue of a graph, the following theorem delivers yet another lower bound on the edge-weighted energy $E_{w}$ of graphs.

Theorem 3.6. If the eigenvalues of $A_{w}(\Gamma)$ are non-zero, then

where $|\lambda_{1}^{w}|$ and $|\lambda_{\nu}^{w}|$ are the maximum and minimum of $|\lambda_{k}^{w}|$.

Proof. Let $|\lambda_{1}^{w}|\geq |\lambda_{2}^{w}|\geq \ldots \geq |\lambda_{\nu}^{w}|$ be the eigenvalues of $A_{w}(\Gamma)$. By putting $r_{k} = |\lambda_{k}|$ and $s_{k} = 1$ in Lemma 2.7, we obtain

Hence the proof is completed. □

The following two results deliver lower bounds on $E_w$ in terms of $EM_1$, the smallest and largest eigenvalues of graphs.

Theorem 3.7. Assuming $\lambda_{1}^{w}\geq \lambda_{2}^{w}\geq \ldots \geq \lambda_{\nu}^{w}$ to be the eigenvalues of $A_{w}(\Gamma)$, where $\Gamma$ is an $(\nu, m)$-graph. Then

Proof. Let $\lambda_{1}^{w}\geq \lambda_{2}^{w}\geq \ldots \geq \lambda_{\nu}^{w}$ be the eigenvalues of $A_{w}(\Gamma)$. By putting $r_{k} = |\lambda_{k}^{w}|$, $s_{k} = 1$, $p = \lambda_{\nu}^{w}$ and $P = \lambda_{1}^{w}$, in Lemma 2.9, we obtain

Hence, the proof has been completed. □

Theorem 3.8. Assuming $\lambda_{1}^{w}\geq \lambda_{2}^{w}\geq \ldots \geq \lambda_{\nu}^{w}$ to be the eigenvalues of $A_{w}(\Gamma)$, where $\Gamma$ is an $(\nu, m)$-graph, then

Proof. Let $\lambda_{1}^{w}\geq \lambda_{2}^{w}\geq \ldots \geq \lambda_{\nu}^{w}$ be the eigenvalues of $A_{w}(\Gamma)$. By putting $r_{k} = |\lambda_{k}^{w}|$, $s_{k} = 1$, $p = \lambda_{\nu}^{w}$ and $P = \lambda_{1}^{w}$, in Lemma 2.9, we obtain

Hence the proof has been furnished. □

Next, a sharp upper extremal value on $E_w$ is proven.

Theorem 3.9. If $\Gamma$ is a non-empty graph of order $\nu$. Then

Proof. Let $\lambda_{1}^{w}\geq \lambda_{2}^{w}\geq \ldots \geq \lambda_{\nu}^{w}$ be the eigenvalues of $A_{w}(\Gamma)$. Substituting $p_{k} = |\lambda_{k}^{w}| = q_{k}$ and $r_{k} = s_{k} = m_{k} = \nu_{k} = 1$ in Lemma 2.5, we obtain

□

4.   Applications on the edge-weighted energy of graphs

Until now, many researchers have studied the predictive potential of molecular descriptors (mainly degree-, distance-, and eigenvalue-based) for estimating the $\pi$-electron energy ($E_{\pi}$) of benzenoid hydrocarbons (BHs) and also for estimating the enthalpy of formation $\Delta H^o_f$ and boiling point $B_p$ of BHs. For instance, Hayat et al. ^[16] (resp. Hayat and coauthors ^[14,27]) investigated the efficiency of degree-dependent (resp. eigenvalues-dependent) graphical descriptors for estimating $E_{\pi}$ of BHs. Similar studies were conducted by Hayat et al. ^[15] (Hayat and Liu ^[12]) for distance-related and temperature-based graphical descriptors. For predicting physicochemical properties such as $B_p$ and $\Delta H^o_f$ of BHs, comparative studies for distance-dependent, temperature-related, degree-related, and eigenvalue-related descriptors were conducted in ^{[13,17,18,26]}, respectively. For more studies on QSPR, we refer to ^[1,11,29,32].

In this section, we calculate energy and edge-weighted energy for molecular graphs of 22 benzenoid hydrocarbons which are listed in Table 1. The adjacency matrix is defined, and eigenvalues, energy, and edge-weighted energy are calculated using the Python programming language. The correlation and regression models are obtained for the physicochemical properties of 22 BHs, for which the data is taken from ^[25] (refer to Table 2), and the predictive ability is tested using energy $E(\Gamma)$ and edge-weighted energy $E_{w}(\Gamma)$. In Table 3, the values of 11 molecular descriptors for 22 BHs are listed, which are from ^[25]. Note that we consider carbon–carbon structures as chemical graphs. One can consider other construction, of chemical graph based on molecular alignment ^[22].

This subsection records all the data sets that we employ for our structure-property models.

5.   Results and discussions

The intercorrelation between the physicochemical properties of polycyclic aromatic hydrocarbons, such as Kovats retention index (RI), acentric factor ($\omega$), octanol-water partition coefficient ($\log P$), boiling point (BP), enthalpy of formation ($\Delta H_{f}$) and entropy ($S$), with graph energy $E(\Gamma)$ and edge-weighted energy $E_{w}(\Gamma)$, is analysed in Table 4. Also, the intercorrelation between 11 molecular descriptors such as atom bond connectivity index (ABC), forgotten index (F), $1^{\mathrm{st}}$ and $2^{\mathrm{nd}}$ Zagreb invariants ($M_1$ and $M_2$), sum division degree index (SDD), reciprocal Randi$\acute{c}$ index (RR), classical Randi$ć$ index (R), redefined Zagreb index (ReZM) with graph energy $E(\Gamma)$ and edge-weighted energy $E_{w}(\Gamma)$ is listed in Table 5. We observe that the 11 molecular descriptors are highly intercorrelated with edge-weighted energy $E_{w}(\Gamma)$ with $\text{r} > 0.97$, which is highlighted in Table 5.

The value of $\text{r}$ for $E_{w}(\Gamma)$ ranges from 0.972 to 0.980.

5.1.   Regression models

The quadratic regression models for physico-chemical properties (PPs) such as Kovats retention index (RI), acentric factor ($\omega$), octanol–water partition coefficient ($\log P$), boiling point (BP), enthalpy of formation ($\Delta H_{f}$), and entropy ($S$) are derived with respect to graph energy $E(\Gamma)$ and edge-weighted energy $E_{w}(\Gamma)$. The symbols $\nu$, $\text{r}$, $F$, and $SE$ are used to represent population, correlation coefficient, $F$-values, and the standard error of the estimate, respectively. Note that, in general, quadratic models have very bad predictive power, even if they have good estimating power. The reader is referred to ^[21] for diversity in detailed regression analysis.

The quadratic regression model is defined as

The quadratic regression of PP with $E(\Gamma)$ is as follows:

The quadratic regression of PP with $E_{w}(\Gamma)$ is as follows:

5.2.   Analysis

The following analysis can be made from the quadratic regression models:

● The correlation coefficient $\text{r}$ for quadratic regression models gives high predictability for physicochemical properties with respect to graph energy $E(\Gamma)$ and edge-weighted energy $E_{w}(\Gamma)$.

● The quadratic regression models for $E(\Gamma)$ give high intercorrelation with a correlation value of $\text{r} = 0.9823$ for the acentric factor.

● The degree-2 polynomial regression model for $E(\Gamma)$ gives appreciable intercorrelation with a correlation value of $\text{r} = 0.9230$ for the boiling point, $\text{r} = 0.9241$ for the log P, $\text{r} = 0.9428$ for the retention index.

● The quadratic regression model for $E(\Gamma)$ is weakly intercorrelation with correlation value $\text{r} = 0.8491$ for the entropy, and $\text{r} = 0.8831$ for the enthalpy.

● The quadratic regression models for $E_{w}(\Gamma)$ give high intercorrelation with a correlation value of $\text{r} = 0.9705$ for the boiling point and $\text{r} = 0.9741$ for retention index.

● The quadratic regression model for $E_{w}(\Gamma)$ gives appreciable intercorrelation with a correlation value of $\text{r} = 0.9208$ for entropy, $\text{r} = 0.9544$ for acentfac, $\text{r} = 0.9695$ for log P, and $\text{r} = 0.9386$ for the enthalpy.

● From all the 12 quadratic regression models, it has been observed that the significance $F$ is 0.000.

The scattered curve diagram of $E(\Gamma)$ with physiochemical properties are depicted in Figures 1–6.

The scattered curve diagram of $E_{w}(\Gamma)$ with physicochemical characteristics are depicted in Figures 7–12.

6.   Conclusions

This paper puts forward the edge-weighted adjacency matrix $A_{w}(\Gamma)$ of a graphical structure $\Gamma$. The energy $E_{w}(\Gamma)$ as well as the spectral radius $\lambda_{1}^{w}$ of the $A_{w}(\Gamma)$ have been studied, and lower and upper extremes are derived for $\lambda_{1}^{w}$ and $E_{w}(\Gamma)$ in terms of other graphical parameters. Further, we calculated the graph energy and edge-weighted energy of 22 BHs by drawing their molecular graphs to check the predictive potential of the physicochemical characteristics of BHs. Polynomials of degree-2 regression models were generated for Kovats retention index (RI), acentric factor ($\omega$), octanol-water partition coefficient ($\log P$), boiling point (BP), enthalpy of formation ($\Delta H_{f}$), and entropy ($S$) using theses two graph energies. We also found correlation coefficients of the physicochemical properties and molecular descriptors of BHa corresponding to the two graph energies $E(\Gamma)$ and $E_{w}(\Gamma)$.

Author contributions

All authors contributed equally to this paper. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

S. Hayat is supported by UBD Faculty Research Grants under Grant Number UBD/RSCH/1.4/FICBF(b)/2022/053 and the National Natural Science Foundation of China (Grant No. 622260-101). A. Khan was sponsored by the National Natural Science Foundation of China grant Nos. 622260-101 and 12250410247, and also by the Ministry of Science and Technology of China, grant No. WGXZ2023054L. Mohammed J. F. Alenazi extends his appreciation to Researcher Supporting Project number (RSPD2024R582), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that they have no known competing financial interests.