The Sombor index of trees and unicyclic graphs with given maximum degree

Let $d_G(v)$ be the degree of the vertex $v$ in a graph $G$. The Sombor index of $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, which is a new degree-based topological index introduced by Gutman. Let $\mathscr{T}_{n,\Delta}$ and $\mathscr{U}_{n,\Delta}$ be the set of trees and unicyclic graphs with $n$ vertices and maximum degree $\Delta$, respectively. In this paper, the tree and the unicyclic graph with minimum Sombor index among $\mathscr{T}_{n,\Delta}$ and $\mathscr{U}_{n,\Delta}$ are characterized.


Introduction
Let G be a simple undirected graph with vertex set V (G) and edge set E(G). unicyclic graph obtained by attaching 2∆ − n + 1 pendant vertices and n − ∆ − 1 paths P 2 to one vertex of a cycle C 3 .
A spider is a tree with at most one vertex of degree more than two, the unique vertex called the hub of the spider. A leg of a spider is a path from the hub to one of the leaves. Let S(a 1 , a 2 , . . . , a k ) be a spider with k legs P 1 , P 2 , . . . , P k for which the lengths l(P i ) = a i for 1 ≤ i ≤ k. Note that T n, ∆ is also a spider. For convenience, denote by T ∆ a spider with length of all ∆ legs greater than 2, shown in Fig. 1.1. Let U ∆ be a unicyclic graph obtained by attaching ∆ − 2 paths of length at least 2 to a cycle, shown in Fig. 1.1.
, which is a novel vertex-degree-based molecular structure descriptor proposed by Gutman [4]. Redžepović [13] showed that the Sombor index may be used successfully on modeling thermodynamic properties of compounds due to the fact that the Sombor index has satisfactory prediction potential in modeling entropy and enthalpy of vaporization of alkanes. Das et al. [2] and Wang et al. [15] obtained the relations between the Sombor index and some other well-known degree-based descriptors, such as the first Zagreb index, the second Zagreb index, the forgotten topological index and so on. For other related results, one may refer to [5,6,7,8,12] and the references therein.
The extremal value problem of the topological index is of interest in mathematical chemistry. The investigation of extremal value of the Sombor index of graphs has quickly received much attention. Gutman [4] obtained extremal values of the Sombor index among the set of (connected) graphs and the set of trees. Cruz et al. [1] studied the Sombor index of chemical graphs, and characterized the graphs extremal with respect to the Sombor index over the following sets: chemical graphs, chemical trees, and hexagonal systems.
Deng et al. [3] obtained a sharp upper bound for the Sombor index among all molecular trees with fixed numbers of vertices, and characterized those molecular trees achieving the extremal value. Liu [10] determined the first fourteen minimum chemical trees, the first four minimum chemical unicyclic graphs, the first three minimum chemical bicyclic graphs, the first seven minimum chemical tricyclic graphs. Réti et al. [14] characterized graphs with the maximum Sombor index in the classes of all connected unicyclic, bicyclic, tricyclic, tetracyclic, and pentacyclic graphs of a fixed order. Lin et al. [11] obtained lower and upper bounds on the spectral radius, energy and Estrada index of the Sombor matrix of graphs, and characterized the respective extremal graphs.
The purpose of this paper is to study the extremal value problem of Sombor index of trees and unicyclic graphs with given maximum degree. The following theorems are showed.
with equality if and only if T ∼ = T ∆ .
with equality if and only if T ∼ = T n, ∆ .
Corollary 1.2 Let T be a chemical tree with n ≥ 7 vertices. If ∆ = 3 or 4, then with equality if and only if U ∼ = U ∆ .
with equality if and only if U ∼ = U n, ∆ .
with equality if and only if T ∼ = U 3 or T ∼ = U 4 .

Preliminaries
The distance d G (u, v) between two vertices u, v of G is the length of one of the shortest (u, v)-path in G. Denote by G the complement of G and by P uv the path between vertices u and v. Let G − u denote the graph that arises from a graph G by deleting the vertex u ∈ V (G) and all the edges incident with u. Let G − uv denote the graph that arises from G by deleting the edge uv ∈ E(G). Similarly, G + uv is the graph that arises from G by Let a 1 ≥ a 2 ≥ . . . ≥ a n and b 1 ≥ b 2 ≥ . . . ≥ b n be such elements in U that inequalities a 1 + a 2 + . . . + a n ≥ b 1 + b 2 + . . . + b i hold for every i ∈ {1, 2, . . . , n} and equality holds  Proof. From given conditions, Combining the above arguments, we have the proof. ✷ 3 The proof of Theorem 1.1 In this section, we determine the trees with the minimum Sombor index among n-vertex trees with maximum degree ∆. When ∆ = 2, T n,∆ = {P n } and when ∆ = n − 1, T n,∆ = {S n }. From now on, we assume 3 ≤ ∆ ≤ n − 2.
Proof of Theorem 1.1 Let T be an n-vertex tree with maximum degree ∆ that minimize the Sombor index and let v 0 be a ∆-vertex of T . We will show the following Claims 1-3, which, put together, will get our proof.
Then we can get a new tree T 1 ∈ T n,∆ by running graph transformation A 1 on v. By Lemma 2.2, SO(T 1 ) < SO(T ), which contradicts the choice of T . Thus T is a spider. ✷ Let T = S(a 1 , a 2 , . . . , a ∆ ) with ∆ legs P 1 , P 2 , . . . , P ∆ , and the lengths l(P i ) = a i for 1 ≤ i ≤ ∆. Without loss of generality, we assume a 1 ≥ a 2 ≥ . . . ≥ a ∆ .
Proof of Theorem 1.3 Let U be an n-vertex unicyclic graph with maximum degree ∆ that minimize the Sombor index. Suppose C is the unique cycle of U. If there exist a ∆-vertex on C, then we chose it and denote by v 0 ; otherwise, chose any ∆-vertex, also denote by v 0 . First, we assume v 0 / ∈ V (C). Then there is a vertex v ∈ V (C) such that We will show the following Claims 1-5, which, put together, will get our proof.
Proof. If the claim is not true, there are three cases: And let v 0 v 1 v 2 . . . v t be one of the paths from v 0 to pendant vertex v t , where t ≥ 1 and Claim 3. v = v 0 , that is to say, there must be v 0 ∈ V (C).
Proof. For otherwise, we can get a new unicyclic graph We have now in U 3 , d U 3 (v) = ∆ and d U 3 (v 0 ) = 3, then there are at least two paths starting from v 0 to pendant vertices of U 3 , similarly, by running transformation A 1 on v 0 , we can get a contradiction, which contradicts the choice of U. Thus v 0 ∈ V (C). ✷ By Claims 1-3, U is a unicyclic graph obtained by attaching and ∆−2 paths to a cycle C. Let v 1 , v 2 ∈ V (C). Similar to the proof of Theorem 1.1, denote by P i the path from v 0 to a pendant vertex of U and v i ∈ P i , where 3 ≤ i ≤ ∆. Without loss of generality, we can assume that l(P 3 ) ≥ l(P 4 ) ≥ . . . ≥ l(P ∆ ).
This completes the proof of Theorem 1.3. ✷