Extreme graphs on the Sombor indices

1.   Introduction

For a graph $F$ with vertex set $V(F)$ and edge set $E(F)$, we let $e(F)$, $\delta(F)$, $\Delta(F)$ and $\overline{F}$ denote the number of edges, minimum degree, maximum degree, and complement of $F$, respectively.

Throught this paper, $F = (V(F), E(F))$ denotes a finite simple graph, it has no loops and parallel edges in graph $F$, for a nonempty subset $S$ of $F$, the subgraph $F[S]$ of $F$ induced by $S$, more generally, the graph $F-S$ is the induced subgraph $F[V-S]$ of $F$, the complement $\overline{F}$ of a graph $F$ is that graph with vertex set $V(F)$ such that no vertices are adjacent in $\overline{F}$ if and only if these vertices are not adjacent in $F$. We refer the readers to the book ^[1] for undefined notations and terminologies.

Recently, a new index, introduced by Gutman ^[16], is defined as

where $\deg_F(v_i)$ is the degree of the vertex $v_i$ in $F$. This new parameter is called Sombor index. Gutman ^[16] also proposed another concept

and this new parameter is called reduced Sombor index.

Some bounds for $\operatorname{SO}(F)$ of trees and general graphs are presented in ^[16] and some properties are established. Deng et al. ^[13] gave an upper bound for all chemical trees with fixed order, and characterized extremal trees. Redžepović ^[21] examined the potentials of (reduced, average) Sombor index. Cruz et al. ^[2] characterized the extremal graphs on the Sombor index for chemical graphs, chemical trees, and hexagonal systems. In ^[6,8,12], Das et al. gave some bounds for Sombor index in terms of different graph parameters. In ^[7,22], the authors characterized the maximal graph of Sombor index among all connected $\nu$-cyclic graphs of order $n$, where $0\leq \nu \leq n-2$.

For an integer $n \geq 2$, $P_n$ is a path of order $n$ with $n-1$ edges. A chemical tree (or molecular tree) is a tree with degree at most $4$. A graph is called molecular graph if its maximum degree is at most four.

Deng et al. ^[13] proposed the following problems.

Problem 1. ^[13] Which molecular trees with perfect matching reach the maximum value of $\operatorname{SO}$ and $\operatorname{SO_{red}}$?

Problem 2. ^[13] Which molecular graphs reach the maximum value of $\operatorname{SO}$ and $\operatorname{SO_{red}}$?

Problem 3. ^[13] Which connected graphs of order $n$ and size $m$ reach the extremal (minimum and maximum) of the (reduced) Sombor index?

Cruz and Rada ^[3] partly solved Problem 2. But the others remain open. Gutman ^[16] derived the following bounds for Sombor index.

Theorem 1.1. ^[16] Let $F$ be a graph of order $n$. Then

with equality if and only if $F\cong \overline{K_n}$ or $F \cong K_n$. Recall that $\operatorname{SO}(\overline{K_n}) = 0$ and $\operatorname{SO}(K_n) = \frac{n(n-1)^2}{\sqrt{2}}$.

Theorem 1.2. ^[16] Let $T$ be a tree of order $n$. Then

with equality if and only if $T\cong P_n$ or $T \cong S_n$, where $S_n$ is a star with $n$ vertices. Recall that $\operatorname{SO}(S_n) = (n-1)\sqrt{n^2-2n+2}$ and $\operatorname{SO}(P_n) = 2\sqrt{5}+2(n-3)\sqrt{2}$.

This paper is scheduled as follows. In Section $2$, we determine the extreme graphs with the maximum value of Sombor index and the extremal connected graphs with the (reduced) Sombor index. In Section $3$, some relations among the chemistry indexes are presented. In Section $4$, we conclude this paper.

2.   Extreme molecular graphs for $\operatorname{SO}$ and $\operatorname{SO_{red}}$

Here are some properties of (reduced) Sombor index.

Theorem 2.1. ^[13] Let $F$ be a graph with $n$ vertices. Then

with equality if and only if $F\cong tK_2 \cup (n-2)K_1$ or $F\cong K_n$. Recall that $\operatorname{SO}_{red}(tK_2 \cup (n-2t)K_1) = 0,$ $0\leq t \leq \lfloor \frac{n}{2}\rfloor$ and $\operatorname{SO}(K_n) = \frac{\sqrt{2}}{2}n(n-1)(n-2)$.

Let $\mathcal{C}G_n$ be the set of molecular graphs with $n$ vertices.

Theorem 2.2. ^[2] Let $F\in \mathcal{C}G_n$, $n\ge 5$. Then

with equality if and only if $F\cong P_n$ or $F$ is a $4$-regular graph of order $n$.

Lemma 2.1. For any graph $F$, we have

Proof. Suppose that $F$ is connected. Consider the edge weight sum by double counting method for every edge $v_iv_j\in E(F)$. The weight sum that the vertex $v_i$ contributes to its adjacent edges is $d_F(v_i)^2$. On the other hand, let us calculate the edge weight sum by counting each edge, this implies that we have completed the proof of Eq (2.1).

Suppose that $F$ is a disconnected graph. Let $F = H_1\cup H_2\cup \cdots\cup H_p$, where $H_k$ $(k = 1, \, 2, \ldots, \, p)$ is the $k$-th connected component in $F$. From the above result, we conclude that

Liu et al. has solved Problem 2 in ^[24].

Theorem 2.3. ^[24] Let $F\in \mathcal{C}G_n$. Then

The equality holds if and only if $H$ is a $4$-regular graph of order $n$.

Let $\mathcal{C}G^k_n$ be the set of connected graphs with $n$ vertices such that the degree of each vertex is at most $k$. we calculate the maximum value of Sombor index in $\mathcal{C}G^k_n$. We now give an bound on $\mathcal{C}G^k_n$ about (reduced) Sombor index as below.

Theorem 2.4. Let $F\in \mathcal{C}G^k_n$. Then

The left (right, respectively) equality holds if and only if $F\cong P_n$ ($F\cong G_k$, respectively), where $G_k$ stands for a $k$-regular graph of order $n$. Recall that $\operatorname{SO}_{red}(G_k) = \frac{\sqrt{2}}{2}\, nk(k-1)$ and $\operatorname{SO}_{red}(P_n) = (n-3)\sqrt{2}+2$.

Proof. For any graph $F$ (with no isolated vertices), let $s_i = s_i(F)$ be the number of vertices of degree $i$ in $F$, and let $e_{i, j} = e_{i, j}(F)$ be the number of edges in $F$ connecting the vertices of degrees $i$ and $j$.

It means that

It is well-known that the following relations hold: for two integers $i, j\in\{1, 2, \cdots, k\}$,

where

Let $\Upsilon = \{(i, j)|(i, j)\in \mathbb{N} \times \mathbb{N}, 1\leq i\leq j\leq k\}$ and $\Omega = \{(i, j)|(i, j)\in \Upsilon, (i, j)\neq (k, k) \}$. By (2.2) and (2.3), we have

Note that the value of reduced Sombor index of molecular graphs with $n$ vertex is equivalent to

Thus,

Let $h(i, j) = \sqrt{(i-1)^2+(j-1)^2}-\frac{ k(k-1)}{2}\sqrt{2}(\frac{1}{i}+\frac{1}{j})$. where $(i, j) \in \Omega$, it is easy to see that $h(k, k) = 0$ and $h(i, j) < 0$ for $(i, j) \in \Omega$, and then

we have

It implies that $e(i, j) = 0$ for all $(i, j)\in \Omega$, in the sake of reaching the maximum value of $\operatorname{SO_{red}}(F)$. Then $F$ is a $k$-regular graph. Conversely, if $F$ is a $k$-regular graph, then $\operatorname{SO_{red}}(F) = \frac{ k(k-1)n}{2}\sqrt{2}$. Recall that $F$ is an extremal graph which is a $k$-regular graph. For lower bound, it is similarly to the proof of Theorem 1.2, so, we give the result without proof. Similarly to the proof of Theorem 2.4, we have a result as below.

Corollary 2.1. Let $F\in \mathcal{C}G^k_n$. Then

with left (right, respectively) equality holds if and only if $F\cong P_n$ ($F\cong G_k$, respectively), where $G_k$ stands for a $k$-regular graph of order $n$. Recall that $\operatorname{SO}(G_k) = \frac{\sqrt{2}}{2}\, nk^2$ and $\operatorname{SO}(P_n) = 2\sqrt{5}+2(n-3)\sqrt{2}$.

Let $\mathcal{C}G_{n, m}$ be the class of graphs with $n$ vertices and $m$ edges. We now consider Problem 3. Let us calculate the minimum value of Sombor index for $F \in \mathcal{C}G_{n, m}$.

Theorem 2.5. Let $F \in \mathcal{C}G_{n, m}$.

$(i)$

with equality if and only if $F$ is regular with order $n$.

$(ii)$ If $F$ is triangle-free, then

with equality if and only if $F \cong \overline{K_n}$.

Proof. The following fact is immediate.

Fact 1. For $\omega_{1}, \omega_{2} \in \mathbb{R^{+}}$, we have

Moreover, the left (right) equality holds if $\omega_{1}\omega_{2} = 0$ $(\omega_{1} = \omega_{2})$.

$(i)$ By Lemma 2.1 and Cauchy-Schwarz inequality, it follows that

The right equality holds if $d_F(v_1) = d_F(v_2) = \cdots = d_F(v_n)$. Using the above result with Handshaking theorem, we have

it admits that

where equality holds if $d_F(v_1) = d_F(v_2) = \cdots = d_F(v_n)$. We take $\omega_{1} = d_F(v_i)$ and $\omega_{2} = d_F(v_j)$ in Fact 1, and the right inequality (2.5) will become

by (2.6). Moreover, the equality holds if $F$ is a regular graph of order $n$.

$(ii)$ For $v_iv_j\in E(F)$, since $F$ is triangle-free, it follows that $v_i$ and $v_j$ cannot have a common neighbor, and hence $d_F(v_i)+ d_F(v_j) \leq n$. Summing this inequality over all the edges, we have

By Lemma 2.1, we have

For $\omega_{1} = d_F(v_i)$ and $\omega_{2} = d_F(v_j)$, the left inequality (2.5) is transformed into

by (2.7). In addition, the equality holds if $d_{F}(v_i)\, d_{F}(v_j) = 0$ for $v_iv_j\in E(F)$, $i.e.$, $F \cong \overline{K_n}$.

Theorem 2.6. Let $F \in \mathcal{C}G_{n, m}$.

$(i)$

with equality if $F$ is regular with $|V(F)| = n$.

$(ii)$ If $F$ is triangle-free, then

with equality if $F$ is isomorphic to $S_n$.

Proof. The proof is very similar to Theorem 2.5.

Fact 2. For $\tau_{1}\geq 1$ and $\tau_{2}\geq 1$, $\tau_{1}, \tau_{2}\in \mathbb{R^{+}}$ the following inequality holds:

Moreover, the left (right) equality holds if $(\tau_{1}-1)(\tau_{2}-1) = 0$ $(\tau_{1} = \tau_{2})$.

$(i)$ For $\tau_{1} = d_F(v_i)$ and $\tau_{2} = d_F(v_j)$, according to the above result, it is easy to see that

by (2.6). Moreover, the equality holds if and only if $F$ is a regular graph of order $n$.

$(ii)$ Since $F$ is triangle-free, one can easily see that $(d_F(v_i)-1)+(d_F(v_j)-1)\leq n-2$ for $v_iv_j\in E(F)$. Summing this inequality over all the edges, we obtain

For $\tau_{1} = d(v_i)$ and $\tau_{2} = d(v_j)$, it follows from (2.8) that

Moreover, the equality holds if and only if $d_{F}(v_i) = 1$ or $d_{F}(v_j) = 1$ for $v_iv_j\in E(F)$, and $d_{F}(v_i)+d_{F}(v_j) = n$ for $v_iv_j\in E(F)$, that is, if and only if $F \cong S_n$.

Theorem 2.7. Let $F \in \mathcal{C}G_{n, m}$. Then

with equality if $F\cong K_n$.

Proof. Since $\Delta\leq n-1$, it follows that

for $v_iv_j\in E(F)$. Similarly,

for $v_iv_j\in E(F)$. Then

where equality holds if and only if $d_F(v) = n-1$ for any $v\in V(F)$, $F\cong K_n$. Moreover,

with equality if and only if $F\cong K_n$.

We now give an upper bound on $\operatorname{SO}(F)$ and $\operatorname{SO}_{red}(F)$ in terms of $m$ only.

Proposition 2.1. Let $F$ be a graph with $m$ edges. Then $\operatorname{SO}(F)\leq m(m+1)$ where equality holds if and only if $F\cong \overline{K}_n$. Moreover, $\operatorname{SO}_{red}(F)\leq m^2-m$ where equality holds if and only if $F\cong S_n$.

Proof. For $v_iv_j\in E(F)$, since $d_F(v_i)+d_F(v_j)\leq m+1$, it follows that

Moreover, the equality holds if $F\cong \overline{K}_n$. Therefore,

The equality holds if and only if $F\cong S_n$.

Let $\mathcal{C}G^{\ell}{\{n, m\}}$ be the set of graphs with $n$ vertices and $m$ edges such that $d_F(v_i)+d_F(v_j)\leq \ell$.

Now consider the following question: to determine the maximum value of $\operatorname{SO}(F)$ among all connected graphs of order $n$ with $m$ edges such that $d_F(v_i)+d_F(v_j)\leq \ell$.

Using a proof similar to the Theorem 2.5, we obtain an upper bound.

Corollary 2.2. Let $F \in \mathcal{C}G^{l}{\{n, m\}}$. Then $\operatorname{SO}(F)\leq m\, \ell$.

For $n, \rho_{1}, \rho_{2}, \rho_{3} \in \mathbb{Z^{+}}$ and $\sum_{i = 1}^{i = 3}{\rho_{i}} = n-3$, $U(n, \rho_{1}, \rho_{2}, \rho_{3})$ is a unicyclic graph obtained from a $3$-cycle $C_3$ with $V(C_3) = \{u, v, w\}$ by adding $\rho_{1}$, $\rho_{2}$ and $\rho_{3}$ pendent vertices to the vertices $u$, $v$ and $w$, respectively. $B_{2, 0}(n, n-4, 0, 0)$ stands for two copies of $K_3$ by sharing an edge and connecting the remaining hanging edges to a vertex on the sharing edge.

Cruz and Rada ^[3] characterized the extremal graphs restricted in unicyclic graphs and bicyclic graphs, respectively.

Theorem 2.8. ^[3] If $F$ is the unicyclic graph of order $n \ (n \geq 4)$, then

Theorem 2.9. ^[3] If $F$ is a bicyclic graph of order $n \ (n \geq 6)$, then

For convenience, for a subgraph $H_0$ of $K_n$, denote by $K_n-H_0$ the graph obtained by deleting all edges of $H_0$ from $K_n$. From the structure of $K_n-H_0$, $\overline{K_n-H_0} = \overline{K_{n-|V(H_0)}|}\cup H_0$. In particular, let $H_1(n) = K_n-2K_2$ and $H_2(n) = K_n-P_3$.

Proposition 2.2. Let $F \in \mathcal{C}G_{n, m}$ with $m = \binom{n}{2}-2$ $(n\geq 5)$. Then

where

Proof. Since $F \in \mathcal{C}G_{n, m}$ for $m = \binom{n}{2}-2$, it must satisfy $F\cong H_1(n) = K_n-2K_2$ or $F \cong H_2(n) = K_n-P_3$. For graph $K_n-2K_2$, there are four vertices, say $v_1, v_2, v_3$ and $v_4$, of degree $n-2$. Suppose $v_1v_2$ and $v_3v_4$ denote $2K_2$, then $NE_{\ell_1, \ell_2} = \{uv|deg_F(u) = \ell_1, deg_F(v) = \ell_2\}$ by the definition of Sombor index. It follows that

Similarly, we have

Thus, $\operatorname{SO}(F-P_3)\geq \operatorname{SO}(F-2K_2)$ for $n \geq 5$.

Suppose that $F$ is a connected graph with order $n$ and size $m$. Clearly, $\binom{n}{2} = \frac{n(n-1)}{2} \geq m \geq n-1$. Let $F(m, n)$ be a graph giving the maximum value of the Sombor index. Then $\operatorname{SO}(F)\leq \operatorname{SO}(F(m, n)) = f(m, n)$. If $m = n-1$, then $F$ is a tree. By Theorem 1.2, $\operatorname{SO}(F) \leq (n-1)\sqrt{n^2-2n+2}$ where equality holds if and only if $F \cong S_n$, where $S_n$ is the star of order $n$. Naturally, $f(n-1, n) = (n-1)\sqrt{n^2-2n+2}$ and $F(n-1, n) = S_n$.

If $m = \binom{n}{2} = \frac{n(n-1)}{2}$, then $F$ is a complete graph. By Theorem 1.1, $\operatorname{SO}(F) \leq \frac{n(n-1)^2}{\sqrt{2}}$, where the equality holds if and only if $F\cong K_n$. Then $f(\binom{n}{2}, n) = \frac{n(n-1)^2}{\sqrt{2}}$ and $F(\binom{n}{2}, n) = K_n$.

If $m = \binom{n}{2}-1 = \frac{n^2-n-2}{2}$, then $F$ is $K_n-K_2$, and hence

So,

and

By Theorem 2,

and

We now consider the general situation, for any integer $n, m$, $n-1\leq m \leq \binom{n}{2}$.

Algorithm A1, which gives a graph $F(m, n)$:

Let $1 \leq p \leq n-1$, $s_p = \sum_{i = 1}^{p}{(n-i)}$, $k = \min\{p \, |\, s_p\leq m < s_{p+1}\}$, $t = m-s_k$. Then

Based on the theorem above, we propose the following problem.

Problem 4. Let $F \in \mathcal{C}G_{n, m}$ be a connected graph. Then

with equality if and only if $F \cong F(m, n)$, where $F(m, n)$ isgiven by Algorithm A1, that is, $\operatorname{SO}(F(m, n)) = f(m, n)$.

3.   Relation between $\operatorname{SO}$ and the others indices

The average Sombor index is defined as follows:

The average Sombor index is introduced by Gutman ^[16] in 2021, it is a variant of Sombor index, which is a better measure of the chemical properties of molecular compounds.

Theorem 3.1. Let $K_{\ell_1, \ell_2}$ be a complete bipartite graph of order $n$. Then we have

where

Proof. For $K_{\ell_1, \ell_2}$ $(n = \ell_1+\ell_2, \, \ell_1\leq \ell_2)$, we have $m = \ell_1\, \ell_2 = \ell_1\, (n-\ell_1)$ and

Consider the following function

Then find the derivative of function $f(x)$,

Set $f'(x) = 0$, the set of real roots of this equation is as showing below:

One can easily check that for $1 \leq x \leq \frac{n}{4} \left(2-\sqrt{\frac{1}{2} \left(\sqrt{17}-1\right)}\right)$, it follows that $f'(x) \geq 0$ and for $\frac{n}{4} \left(2-\sqrt{\frac{1}{2} \left(\sqrt{17}-1\right)}\right) \leq x \leq \lfloor \frac{n}{2}\rfloor$, it follows that $f'(x) \leq 0$. So, we know $f(x)$ is an increasing function on $1 \leq x \leq \frac{n}{4} \left(2-\sqrt{\frac{1}{2} \left(\sqrt{17}-1\right)}\right)$ and a decreasing function on $\frac{n}{4} \left(2-\sqrt{\frac{1}{2} \left(\sqrt{17}-1\right)}\right) \leq x \leq\; \lfloor \frac{n}{2}\rfloor$.

Hence,

where $t_1 = \left\lfloor \frac{n}{4} \left(2-\sqrt{\frac{1}{2} \left(\sqrt{17}-1\right)}\right)\right\rfloor$ and $t_2 = \left\lceil \frac{n}{4} \left(2-\sqrt{\frac{1}{2} \left(\sqrt{17}-1\right)}\right)\right\rceil$.

To illustrate the above Theorem 3.1, we give some extremal graphs for $1 \leq n\leq 1000$. For $n \in\; \{2, 3, 4, 5, 6, 7\}$, the extremal graph is $K_{1, n-1}\cong S_{1, n-1}$ and $S_{1, k}$ is a star graph. For $n\geq 8$ and integers $r, t$ with $0 \leq r\leq 15$, if $n-8 = 16t+r$, then the extremal graph $K_{\ell, n-\ell}$ satisfies

Kulli ^[18] proposed the first Banhatti-Sombor index of a connected graph $F$ which is defined as

which is also inspired by Sombor indices.

Theorem 3.2. Let $F$ be a connected graph. Then

with equality if and only if $F$ is a regular graph.

Proof. Since $\delta^{2} \leq d_F(u)d_F(v) \leq \Delta^{2}$, we have

and

Then

Moreover, both equalities hold if and only if $F$ is a regular graph.

Inspired by the definations of the Zagreb indices with applications, Kulli ^[20] introduced the first Gourava index defined as

for the molecular graph $F$, and the Second Gourava index ^[19]

for the molecular graph $F$. The forgotten topological index is defined as ^[17]

We now give an upper bound on $GO_1(F)$ and characterize the extremal graphs.

Theorem 3.3. Let $F$ be a connected graph. Then $GO_1(F)\leq F_1(F)$ with equality if and only if $F\cong P_n$ or $F\cong C_n$.

Proof. Let $uv\in E(F)$. Without loss of generality, we can assume that $d_F(u)\geq d_F(v)$. Also let

If $\Big(d_F(u), \, d_F(v)\Big)\in\{(2, 1), \, (2, 2)\}$, then from (2.8), $A = 0$. Otherwise, $\Big(d_F(u), \, d_F(v)\Big)\notin \{(2, 1), \, (2, 2)\}$. Since $F$ is connected, $d_F(u)\geq 3$. We consider the following two cases:

Case 1. $d_F(u) > d_F(v)$.

If $d_F(u)\geq d_F(v)+2$, then we have

Otherwise, $d_F(u) = d_F(v)+1$. Therefore $d_F(u)\geq 3$ and $d_F(v)\geq 2$. Thus we obtain

Case 2. $d_F(u) = d_F(v)\geq 3$.

Then $\Big(d_F(u)-1\Big)\, \Big(d_F(v)-1\Big)\geq 4$ and hence $A > 0$.

Thus we conclude that $A\geq 0$ with equality if and only if $\Big(d_F(u), \, d_F(v)\Big)\in\{(2, 1), \, (2, 2)\}$. Hence

Moreover, the above equality holds if and only if $F\cong P_n$ or $F\cong C_n$.

Theorem 3.4. Let $F$ be a connected graph. Then

with equality if and only if $F$ is a regular graph.

Proof. By Cauchy-Schwarz inequality, we obtain

with equality if and only if $d^2_F(u)+d^2_F(v) = d^2_F(w)+d^2_F(z)$ for any edges $uv\in E(F)$ and $wz\in E(F)$. Using the above result, we obtain

Moreover, the equality holds in (3.2) if and only if $d_F(u) = \delta$ for all $u\in V(F)$. Hence the equality holds in (3.2) if and only if $F$ is a regular graph.

Theorem 3.5. Let $F$ be a connected graph. Then

with equality if and only if $F$ is a bipartite semiregular graph or $F$ is a regular graph.

Proof. Since

and

by weighted arithmetic-harmonic-mean inequality, we obtain

that is,

Moreover, the equality holds if and only if $d_F(u)d_F(v) = d_F(w)d_F(x)$ for any edges $uv\in E(F)$ and $wx\in E(F)$, that is, if and only if the bipartite semiregular graph or the regular graph.

Kulli ^[19] introduced the sum connectivity Gourava index and the product connectivity Gourava index for a molecular graph $F$, which is defined as

and

respectively.

Theorem 3.6. Let $F$ be a connected graph. Then

with equality if and only if $F$ is a complete graph.

Proof. By inequality between arithmetic and harmonic means for any number $\sqrt{d_F(u)d_F(v)}$ where $uv\in E(F)$, it follows that

Then

with equality if and only if $d_F(u) = d_F(v)$ and $d_F(u)+d_F(v) = 2n-2$.

Using the proof strategy for Theorem 3.6, we have the following similar result.

Theorem 3.7. Let $F$ be a connected triangle-free graph. Then

The eccentricity $\varepsilon_F(v)$ of a vertex $v\in V(F)$ is the distance between $v$ and a vertex farthest from $v$ in $F$. The eccentric connectivity index was introduced by Sharma, Goswami and Madan ^[23] as

For its basic mathematical properties, including lower and upper bounds, see ^{[5,9,10,11,14]} and the references cited therein. The sum of eccentricities of all vertices of $F$ is called the total eccentricity of $F$ ^[4] and denoted by $\zeta(F)$, $\zeta(F) = \sum_{v \in V(F)}\, \varepsilon_F(v)$.

Lemma 3.1. ^[15] Let $F$ be a nontrivial connected graph of order $n$. For each vertex $v \in V(F)$, $d_{F}(v)\leq n-\varepsilon_F(v)$ with equality if and only if $F \cong P_4$ or $F \cong K_n -sK_2$ $(0 \leq s \leq \lfloor \frac{n}{2}\rfloor)$, where $K_n -sK_2$ denotes the graph obtained from the complete graph by removing $s$ independent edges.

Theorem 3.8. Let $F$ be a connected graph of order $n$ with $m$ edges and minimum degree $\delta$. Then

with equality if and only if $F \cong P_4$ or $F$ is isomorphic to an $(n-2)$-regular graph $(n$ is even) or $F\cong K_n$.

Proof. First, we have to prove that for any edge $uv\in E(F)$ $\Big(d_F(u)\geq d_F(v)\Big)$,

that is,

that is,

which is always true. Moreover, the equality holds in (3.6) if and only if $d_F(u) = d_F(v)$. Now,

with equality if and only if $d_{F}(v) = \delta$ for every edge $uv\in E(F)$. Using the above results with Lemma 3.1, we obtain

Both (3.6) and (3.7) equalities hold if and only if $d_F(u) = d_F(v) = \delta$ for any edge $uv\in E(F)$. From the equality in (3.8), we have $F \cong P_4$ or $F \cong K_n -sK_2$ $(0 \leq s \leq \lfloor \frac{n}{2}\rfloor)$ by Lemma 3.1. Hence the equality holds in (3.5) if and only if $F \cong P_4$ or $F$ is isomorphic to an $(n-2)$-regular graph $(n$ is even) or $F\cong K_n$.

Theorem 3.9. Let $F$ be a connected graph of order $n$ with size $m$ and maximum degree $\Delta$. Then

The above equality holds if and only if $F \cong P_4$ or $F$ is isomorphic to an $(n-2)$-regular graph ($n$ is even) or $F\cong K_n$.

Proof. By Cauchy-Schwarz inequality, we obtain

The first part of the proof is done.

Suppose equality holds in (3.9). Then all the inequalities in the above argument occur as equalities. From equality in (3.10), we obtain $d^2_{F}(u)+d^2_{F}(v) = d^2_{F}(w)+d^2_{F}(x)$ for any two edges $uv\in E(F)$ and $wx\in E(F)$, that is, $F$ is a semiregular graph as $F$ is connected. From the equality in (3.11), we have $F \cong P_4$ or $F \cong K_n -sK_2$ $(0 \leq s \leq \lfloor \frac{n}{2}\rfloor)$, by Lemma 3.1. From equality in (3.12), we obtain $d_F(v) = \Delta$ for all $v\in V(F)$ as $d_{F}(v)n > d_{F}(v)\, \varepsilon_F(v)$ with connected graph $F$. From these results, we obtain $F \cong P_4$ or $F$ is isomorphic to an $(n-2)$-regular graph $(n$ is even) or $F\cong K_n$.

Conversely, let $F \cong P_4$. Then $m = n = 4$, $\Delta = 2$, $\zeta^{e}(F) = 16$ and hence

Let $F$ be a graph which is isomorphic to an $(n-2)$-regular graph $(n$ is even). Then $2m = n(n-2)$, $\Delta = n-2$ and $\zeta^{e}(F) = 2n\, (n-2)$. Thus we have

Let $F\cong K_n$. Then $2m = n(n-1)$, $\Delta = n-1$ and $\zeta^{e}(F) = n\, (n-1)$. Thus we have

By the same way, we have the following result and we omit the details proof of it.

Theorem 3.10. Let $F$ be a connected graph of order $n$. Then

with equality if and only if $F \cong P_4$ or $F$ is isomorphic to an $(n-2)$-regular graph ($n$ is even) or $F\cong K_n$.

Proof. Since $d_F(v)\leq \Delta$ for all $v\in V(F)$, from (3.8), we obtain the required result. Moreover, the equality holds if and only if $F \cong P_4$ or $F$ is isomorphic to an $(n-2)$-regular graph $(n$ is even) or $F\cong K_n$.

4.   Conclusions

In this paper, we studied some extremal values on the (reduced) Sombor index of molecular graphs. Furthermore, some relations among the chemistry indexes are presented, in particular, we obtained relationship between Sombor index and the other topological indices, and characterized the extreme graphs. Specifically, we obtained inequalities for Sombor index relating other indices, $i.e.$, the first Banhatti-Sombor index, the first Gourava index, the Second Gourava index, the Sum Connectivity Gourava index, Product Connectivity Gourava index, and Eccentric Connectivity index.

Acknowledgments

The second author was supported by Doctoral research project of Tianjin Normal University (No.52XB2111). The third author is supported by National Research Foundation funded by the Korean government (Grant No.2021R1F1A1050646). The Fourth author was supported by the National Science Foundation of China (Nos.12061059, 11601254, 11551001).

Conflict of interest

The authors declare no conflict of interest.