On the extremal Sombor index of trees with a given diameter

doi:10.1016/j.amc.2021.126731

Applied Mathematics and Computation

Volume 416, 1 March 2022, 126731

https://doi.org/10.1016/j.amc.2021.126731 Get rights and content

Highlights

•
As a new vertex-degree-based topological indices, the Sombor index studied in this paper differs from the earlier vertex-degree-based topological indices because it has a peculiar geometric interpretation.
•
All the $n$ -order trees with diameter 3 are ordered with respect to the Sombor index.
•
The largest and the second largest Sombor indices of $n$ -vertex trees with a given diameter $d \geq 4$ are determined and the corresponding trees are characterized.
•
For $n - d = 3$ , we characterize the extremal $n$ -order trees which reach from the third to the fourth (resp. the sixth, the seventh) largest Sombor indices with $d = 4$ (resp. $d = 5, d \geq 6$ ). For $n - d \geq 4$ , we characterize the extremal $n$ -order trees which reach from the third to the fifth (resp. the eighth, the ninth) largest Sombor indices with $d = 4$ (resp. $d = 5, d \geq 6$ ).
•
As consequences, the top four $n$ -order trees with respect to the Sombor index are characterized.

Abstract

Based on elementary geometry, Gutman proposed a novel graph invariants called the Sombor index $S O (G)$ , which is defined as $S O (G) = \sum_{u v \in E (G)} \sqrt{d_{G}^{2} (u) + d_{G}^{2} (v)},$ where $d_{G} (u)$ and $d_{G} (v)$ denote the degree of $u$ and $v$ in $G$ , respectively. It has been proved that the Sombor index could predict some physicochemical properties. In this paper, we characterize the extremal graphs with respect to the Sombor index among all the $n$ -order trees with a given diameter. Firstly, we order the trees with respect to the Sombor index among the $n$ -vertex trees with diameter 3. Then, we determine the largest and the second largest Sombor indices of $n$ -vertex trees with a given diameter $d \geq 4$ and characterize the corresponding trees. Moreover, for $n - d = 3$ , we characterize the extremal $n$ -order trees which reach from the third to the fourth (resp. the sixth, the seventh) largest Sombor indices with $d = 4$ (resp. $d = 5, d \geq 6$ ). For $n - d \geq 4$ , we characterize the extremal $n$ -order trees which reach from the third to the fifth (resp. the eighth, the ninth) largest Sombor indices with $d = 4$ (resp. $d = 5, d \geq 6$ ). As consequences, the top four $n$ -order trees with respect to the Sombor index are characterized.

Introduction

In this paper, we only consider connected, simple and undirected graphs. As usual, let $P_{n}$ and $S_{n}$ denote the path and the star on $n$ vertices, respectively. For a set $U$ , denote by $| U |$ its cardinality. All the notation and terminology not defined here we refer the reader to Bondy and Murty [1].

Let $G = (V_{G}, E_{G})$ be a graph with vertex set $V_{G}$ and edge set $E_{G}$ . Denote by $n_{G} : = | V_{G} |$ and $m_{G} : = | E_{G} |$ the order and the size of $G$ , respectively. For a vertex $v \in V_{G}$ , the set of its neighbors in $G$ is denoted by $N_{G} (v)$ and its degree in $G$ is defined as $d_{G} (v) : = | N_{G} (v) |$ . If $d_{G} (v) = 1$ , then $v$ is called a pendant vertex (or a leaf) of $G$ and the unique edge incident with a pendant vertex $v$ is called a pendant edge of $G$ . For $E^{0} \subseteq E_{G}$ , let $G - E^{0}$ be the graph obtained from $G$ by deleting all the edges in $E^{0}$ . Similarly, for $E^{0} \cap E_{G} = \emptyset$ , let $G + E^{0}$ be the graph obtained from $G$ by adding all the edges in $E^{0}$ . Specially, if $E^{0} = {u v}$ , then we denote $G - u v : = G - {u v}$ and $G + u v : = G + {u v}$ for short, respectively.

For ${u, v} \subseteq V_{G},$ a $u - v$ path, written as $P_{G} (u, v)$ , is a path connecting $u$ and $v$ in $G$ . The distance between $u$ and $v$ in $G$ , denoted by $d_{G} (u, v)$ , is the length of a shortest path connecting them. The diameter of a graph $G$ is defined as $diam (G) = \max_{{u, v} \subseteq V_{G}} {d_{G} (u, v)}$ . A $u, v$ -path is called a diametrical path of $G$ if $m_{P_{G} (u, v)} = d_{G} (u, v) = diam (G) .$

Topological indices, also called the graph invariants, are often used to study some properties of molecule graphs and characterize their structure. Back in the 1970s, the first and the second Zagreb indices $M_{1}$ and $M_{2}$ were introduced in [15], [16] and defined as $M_{1} = M_{1} (G) = \sum_{v \in V_{G}} d_{G}^{2} (v) = \sum_{u v \in E_{G}} (d_{G} (u) + d_{G} (v)), M_{2} = M_{2} (G) = \sum_{u v \in E_{G}} d_{G} (u) d_{G} (v),$ where $d_{G} (u)$ and $d_{G} (v)$ denote the degree of $u$ and $v$ in $G$ , respectively. Then, a variety of the vertex-degree-based graph invariants have been introduced and were extensively studied, such as the first and the second hyper-Zagreb index, the Randić index, the forgotten index, the inverse degree, the general sum-connectivity index, the geometric-arithmetic index and the arithmetic-geometric index.

Usually, topological indices play significant roles in mathematical chemistry especially in the QSPR/QSAR investigations. One of the most general problems in graph theory is to find out the extremal values regarding to some graph invariant and characterize all the corresponding extremal graphs. For recent advances we refer to [4], [5], [12], [18], [19], [24] and so on.

Among the vertex-degree-based graph invariants, the forgotten index [11], [15] and the reciprocal sum-connectivity index [25] were defined as $F (G) = \sum_{u v \in E_{G}} (d_{G}^{2} (v) + d_{G}^{2} (v)), χ_{\frac{1}{2}} = \sum_{u v \in E_{G}} \sqrt{d_{G} (u) + d_{G} (v)} .$ Mainly based on them, very recently, Gutman [13] defined the Sombor index as follows: $S O (G) = \sum_{u v \in E (G)} \sqrt{d_{G}^{2} (u) + d_{G}^{2} (v)} .$

In [13], [14], Gutman also presented some properties of the Sombor index and characterize the maximal and minimal graphs with respect to the Sombor index over the set of simple graphs, simple and connected graphs and trees. By using some graph parameters, Das et al. [7] gave some lower and upper bounds on the Sombor index of graphs and obtain several relations between the Sombor index and the first and second Zagreb indices of graphs. Wang et al. [23] establish some relations between the Sombor index and some other well-known vertex-degree-based indices of graphs.

Deng, Tang and Wu obtained a sharp upper bound for the Sombor index among all molecular trees with fixed number of vertices. Meanwhile, the chemical importance of the Sombor index was investigated and it was shown that this new index may be useful in predicting physicochemical properties with high accuracy compared to some well-established and often used indices, which can be seen in [9]. Cruz et al. [3] characterized the extremal graph with respect to the Sombor index over the set of chemical graphs, chemical trees and hexagonal systems. In [6], the extremal values of the Sombor index among the unicyclic graphs (resp. bicyclic graphs) were determined. The graphs with a given clique number (resp. chromatic number, number of pendant vertices) attaining the lower and the upper bounds of the Sombor index were characterized in [8]. More results of Sombor indices can be found in [2], [10], [17], [20], [21], [22].

In this paper, we characterize the extremal graphs with respect to the Sombor index among all the $n$ -vertex trees with a given diameter. For convenience, let $T_{n}^{d}$ be the set of all the $n$ -vertex trees with diameter $d$ and $T_{n}$ be the set of all the $n$ -vertex trees. Obviously, $T_{n} = ⋃_{d = 1}^{n - 1} T_{n}^{d} .$ It is easy to check that $T_{n}^{1} = {P_{2}}$ and $T_{n}^{2} = {S_{n}}$ . So we mainly consider $d \geq 3$ in the whole article. Let $T_{n}^{3} = {S_{a, n - a} | a = 2, \dots, ⌊ \frac{n}{2} ⌋}$ , where $S_{a, n - a}$ is a double star which can be obtained from two stars $S_{a}$ and $S_{n - a}$ by adding an edge between their center vertices.

Our first main result orders all the trees among $T_{n}^{3}$ according to the Sombor index.

Theorem 1.1

Among $T_{n}^{3}$ one has $S O (S_{2, n - 2}) > S O (S_{3, n - 3}) > \dots > S O (S_{⌊ \frac{n}{2} ⌋, ⌈ \frac{n}{2} ⌉}) .$

For $d \geq 4$ and $n \geq d + 1$ , let $T_{n}^{d, i}$ be an $n$ -vertex tree obtained from $P_{d + 1} = v_{0} v_{1} \dots v_{d}$ by adding $n - d - 1$ pendant vertices to $v_{i}$ for $i \in {1, 2, \dots, d - 1}$ as depicted in Fig. 1. Obviously, $T_{n}^{d, 1} ≅ T_{n}^{d, d - 1}$ , we denote $T_{n}^{d} : = T_{n}^{d, 1}$ for short.

Our next main result determines the largest and the second largest Sombor indices of $n$ -vertex trees with diameter $d \geq 4$ . The corresponding extremal trees are also characterized.

Theorem 1.2

Let $T$ be in $T_{n}^{d}$ with $d \geq 4$ . We have the following:

(i)
$S O (T) \leq (n - d) \sqrt{1 + {(n - d + 1)}^{2}} + \sqrt{4 + {(n - d + 1)}^{2}} + (d - 3) \sqrt{8} + \sqrt{5}$ with equality if and only if $T ≅ T_{n}^{d}$ , where $T_{n}^{d} = T_{n}^{d, 1} ≅ T_{n}^{d, d - 1}$ ;
(ii)
If $T \neg ≅ T_{n}^{d}$ , then $S O (T) \leq (n - d - 1) \sqrt{1 + {(n - d + 1)}^{2}} + 2 \sqrt{4 + {(n - d + 1)}^{2}} + (d - 4) \sqrt{8} + 2 \sqrt{5}$ with equality if and only if $T \in {T_{n}^{d, i} : i = 2, 3, \dots, d - 2}$ ,

where the graphs $T_{n}^{d, i}$ for $i = 1, 2, \dots, d - 1$ are depicted in Fig. 1.

For $d \geq 4, n \geq d + 3$ , let $P_{d + 1} = v_{0} v_{1} \dots v_{d}$ . Then, for $i, j \in {1, 2, \dots, d - 1}$ , let $T_{n}^{d, i, j}$ be an $n$ -vertex tree obtained from the path $P_{d + 1}$ by attaching $n - d - 2$ (resp. 1) pendant edges to $v_{i}$ (resp. $v_{j}$ ); see Fig. 2. For convenience, let $H_{n}^{d} : = {T_{n}^{d, i, j} : i, j = 1, 2, \dots, d - 1} .$

In the next two main results, we consider the trees among $T_{n}^{d} ∖ {T_{n}^{d, i} : i = 1, 2, \dots, d - 1}$ . On the one hand, we show that the Sombor index of any tree $T \notin H_{n}^{d}$ is less than those of trees among $H_{n}^{d}$ . On the other hand, we order all the trees among $H_{n}^{d}$ according to the Sombor index.

Theorem 1.3

Let $T$ be in $T_{n}^{d} ∖ {T_{n}^{d, i} : i = 1, 2, \dots, d - 1}$ with $d \geq 4$ and $n - d = 3$ .

(i)
$H_{7}^{4} = {T_{7}^{4, 1, 3}, T_{7}^{4, 1, 2}}$ with $S O (T_{7}^{4, 1, 3}) > S O (T_{7}^{4, 1, 2})$ and $S O (T_{7}^{4, 1, 2}) > S O (T)$ for any tree $T \notin H_{7}^{4}$ ;
(ii)
$H_{8}^{5} = {T_{8}^{5, 1, 4}, T_{8}^{5, 1, 3}, T_{8}^{5, 1, 2}, T_{8}^{5, 2, 3}}$ with $S O (T_{8}^{5, 1, 4}) > S O (T_{8}^{5, 1, 3}) > S O (T_{8}^{5, 1, 2}) > S O (T_{8}^{5, 2, 3})$ and $S O (T_{8}^{5, 2, 3}) > S O (T)$ for any tree $T \notin H_{8}^{5}$ ;
(iii)
If $d \geq 6$ , then $S O (T_{n}^{d, 1, d - 1}) > S O (T_{n}^{d, 1, 3}) = \dots = S O (T_{n}^{d, 1, d - 2}) > S O (T_{n}^{d, 1, 2}) > S O (T_{n}^{d, 2, 4}) = \dots = S O (T_{n}^{d, 2, d - 2}) = \dots = S O (T_{n}^{d, d - 4, d - 2}) > S O (T_{n}^{d, 2, 3}) = \dots = S O (T_{n}^{d, d - 3, d - 2})$ for the trees in $H_{n}^{d}$ and $S O (T_{n}^{d, 2, 3}) > S O (T)$ for any tree $T \notin H_{n}^{d}$ .

Theorem 1.4

Let $T$ be in $T_{n}^{d} ∖ {T_{n}^{d, i} : i = 1, 2, \dots, d - 1}$ with $d \geq 4$ and $n - d \geq 4$ .

(i)
If $d = 4$ , then $S O (T_{n}^{4, 1, 3}) > S O (T_{n}^{4, 1, 2}) > S O (T_{n}^{4, 2, 1})$ for the trees in $H_{n}^{4}$ and $S O (T_{n}^{4, 2, 1}) > S O (T)$ for any tree $T \notin H_{n}^{4}$ ;
(ii)
If $d = 5$ , then $S O (T_{n}^{5, 1, 4}) > S O (T_{n}^{5, 1, 3}) > S O (T_{n}^{5, 3, 1}) > S O (T_{n}^{5, 1, 2}) > S O (T_{n}^{5, 2, 1}) > S O (T_{n}^{5, 2, 3}) = S O (T_{n}^{5, 3, 2})$ for the trees in $H_{n}^{5}$ and $S O (T_{n}^{5, 2, 3}) > S O (T)$ for any tree $T \notin H_{n}^{5}$ ;
(iii)
If $d \geq 6$ , then $S O (T_{n}^{d, 1, d - 1}) > S O (T_{n}^{d, 1, 3}) = \dots = S O (T_{n}^{d, 1, d - 2}) > S O (T_{n}^{d, 3, 1}) = \dots = S O (T_{n}^{d, d - 2, 1}) > S O (T_{n}^{d, 1, 2}) > S O (T_{n}^{d, 2, 1}) > S O (T_{n}^{d, 2, 4}) = \dots = S O (T_{n}^{d, 2, d - 2}) = \dots = S O (T_{n}^{d, d - 4, d - 2}) > S O (T_{n}^{d, 2, 3}) = \dots = S O (T_{n}^{d, d - 3, d - 2})$ for the trees in $H_{n}^{d}$ and $S O (T_{n}^{d, 2, 3}) > S O (T)$ for any tree $T \notin H_{n}^{d}$ .

Recall that $T_{n}$ is the set of all the $n$ -vertex trees. Clearly, $P_{n}$ is the unique tree for $n = 1$ or 2. In our last main result, we consider that $n \geq 3$ and characterize the top four largest trees among $T_{n}$ with respect to the Sombor index.

Theorem 1.5

Let $T$ be in $T_{n} ∖ {S_{n}, S_{2, n - 2}, S_{3, n - 3}, T_{n}^{4}}$ with $n \geq 3$ . Then $S O (T) < S O (T_{n}^{4}) < S O (S_{3, n - 3}) < S O (S_{2, n - 2}) < S O (S_{n}) .$

Section snippets

Some preliminaries

In this section, we give some preliminary lemmas. For convenience, we denote $h (x, y) = \sqrt{x^{2} + y^{2}}$ for $x \geq 1$ , $y \geq 1$ and let $l (x, y) = h (x, y) - h (x - 1, y)$ for $x \geq 2$ , $y \geq 1$ .

Lemma 2.1

[13]

For $x \geq 1, y \geq 1$ , $h (x, y) = h (y, x)$ and $h (x, y)$ is strictly increasing on $x$ (resp. $y$ ).

Lemma 2.2

$h (n - 1, 1) > h (n - 2, 2) > \dots > h (⌈ \frac{n}{2} ⌉, ⌊ \frac{n}{2} ⌋) = h (⌊ \frac{n}{2} ⌋, ⌈ \frac{n}{2} ⌉) < \dots < h (2, n - 2) < h (1, n - 1)$ .

Proof

Let $f (x) = h (x, n - x) = \sqrt{x^{2} + {(n - x)}^{2}}$ , we only need to prove that $f (x)$ is strictly increasing on $x \geq \frac{n}{2}$ while strictly decreasing on $x \leq \frac{n}{2}$ . By direct calculation, the derivative of $f (x)$ is $f^{'} (x) = \frac{1}{2} \cdot \frac{2 x - 2 (n - x)}{\sqrt{x^{2} + {(n - x)}^{2}}} = \frac{2 x - n}{\sqrt{x^{2} + (}}$

Proofs of Theorems 1.1 and 1.2

In this section, we give the proofs of Theorems 1.1 and 1.2.

Proof of Theorem 1.1

Clearly, $T_{n}^{3} = {S_{a, n - a} | a = 2, \dots, ⌊ \frac{n}{2} ⌋}$ , where $S_{a, n - a}$ is a double star obtained from two stars $S_{a}$ and $S_{n - a}$ by adding an edge between their center vertices. In order to complete the proof, it suffices to show $S O (S_{a, n - a}) > S O (S_{a + 1, n - a - 1})$ for $2 \leq a \leq ⌊ \frac{n}{2} ⌋ - 1$ .

Recall that $h (x, y) = \sqrt{x^{2} + y^{2}}$ and $l (x, y) = h (x, y) - h (x - 1, y)$ . By (1.1) one has $\begin{matrix} S O (S_{a + 1, n - a - 1}) - S O (S_{a, n - a}) & = & a h (a + 1, 1) + (n - a - 2) h (n - a - 1, 1) + h (a + 1, n - a - 1) \\ - (a - 1) h (a, 1) - (n - a - 1) h (n - a, 1) - h (a, n - a) \\ = & (a - 1) [h (a + 1, 1) - h (a, 1)] - (n \end{matrix}$

Proofs of Theorems 1.3 and 1.4

In this section, we give the proofs of Theorems 1.3 and 1.4. Recall that $P = v_{0} v_{1} \dots v_{d}$ is a diametrical path of $T$ and $T (v_{i})$ be the component of $T - E_{P}$ with $v_{i} \in T (v_{i})$ for all $i \in {0, 1, 2, \dots, d}$ . Clearly, $d_{T} (v_{0}) = d_{T} (v_{d}) = 1$ and $T (v_{1})$ (resp. $T (v_{d - 1})$ ) is a star with center $v_{1}$ (resp. $v_{d - 1}$ ).

In order to prove our results, we need the following lemmas.

Lemma 4.1

Let $T$ be in $T_{n}^{d}$ satisfying that $S O (T)$ is as large as possible. For every $i \in {2, 3 \dots, d - 2},$ if $T (v_{i})$ is not a star with center $v_{i}$ , then $T (v_{i})$ is a tree obtained from a star $S$

Proof of Theorem 1.5

In this section, we give the proof of Theorem 1.5. That is, we characterize the extremal graphs with respect to the Sombor index among the set of all the $n$ -vertex trees. Clearly, $P_{n}$ is the unique tree for $n = 1$ or 2. We only consider that $n \geq 3$ and it is easy to get that $d \geq 2 .$

Recall that $T_{n}$ is the set of all the $n$ -vertex trees. Clearly, $T_{3} = {S_{3}}$ , our result is obvious. Now we consider $n \geq 4$ . Note that $T_{n}^{2} = {S_{n}}, T_{n}^{3} = {S_{a, n - a} | a = 2, \dots, ⌊ \frac{n}{2} ⌋}$ and by (1.1) one has $\begin{matrix} S O (S_{n}) - S O (S_{2, n - 2}) & = & (n - 1) \sqrt{n^{2} - 2 n + 2} - (n - 3) \sqrt{n^{2} - 4 n + 5} - \sqrt{n^{2} - 4 n +} \end{matrix}$

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: The first author acknowledges the financial support from the National Natural Science Foundation of China (Grant No. 11671164, 12171190) and the third author acknowledges the financial support from the National Natural Science Foundation of China (Grant No. 11901179), the Research ability cultivation fund of HUAS (Grant No. 2020kypytd006) and the Scientific research initial funding of Hubei University of Arts and Science.

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On the extremal Sombor index of trees with a given diameter

Highlights

Abstract

Introduction

Section snippets

Some preliminaries

[13]

Proofs of Theorems 1.1 and 1.2

Proofs of Theorems 1.3 and 1.4

Proof of Theorem 1.5

MATCH Commun. Math. Comput. Chem.

J. Math. Chem.

MATCH Commun. Math. Comput. Chem.

Int. J. Quantum. Chem.

Graph Theory

Extremal values on the Sombor index of trees

MATCH Commun. Math. Comput. Chem.

Sombor index of chemical graphs

Appl. Math. Comput.

Extremal values of vertex-degree-based topological indices of chemical trees

Appl. Math. Comput.

On Sombor index

Symmetry

Some extremal graphs with respect to Sombor index

Mathematics

Molecular trees with extremal values of Sombor indices

Int. J. Quantum Chem.

The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs

Int. J. Quantum. Chem.