On a Conjecture About the Sombor Index of Graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The Sombor and reduced Sombor indices of $G$ are defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{deg_G(u)^2+deg_G(v)^2}$ and $SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(deg_G(u)-1)^2+(deg_G(v)-1)^2}$, respectively. We denote by $H_{n,\nu}$ the graph constructed from the star $S_n$ by adding $\nu$ edge(s) $(0\leq \nu\leq n-2)$, between a fixed pendent vertex and $\nu$ other pendent vertices. R\'eti et al. [T. R\'eti, T. Do\v{s}li\'c and A. Ali, On the Sombor index of graphs, $\textit{Contrib. Math. }$ $\textbf{3}$ (2021) 11-18] proposed a conjecture that the graph $H_{n,\nu}$ has the maximum Sombor index among all connected $\nu$-cyclic graphs of order $n$, where $5\leq \nu \leq n-2$. In this paper we confirm that the former conjecture is true. It is also shown that this conjecture is valid for the reduced Sombor index. The relationship between Sombor, reduced Sombor and first Zagreb indices of graph is also investigated.


Basic Definitions
Throughout this paper, all graphs considered are finite, undirected and simple. Let G be such a graph with vertex set V = V (G) and edge set E = E(G). We recall that the We also use the notation m i,j (G) for the number of edges of G with endpoints of degrees i and j.
A graph G with this property that the degree of each vertex is at most four is called a chemical graph. Suppose V (G) = {v 1 , v 2 , . . . , v n } and d G (v 1 ) ≥ d G (v 2 ) ≥ · · · ≥ d G (v n ).
Then the sequence d(G) = (d G (v 1 ), d G (v 2 ), . . . , d G (v n )) is called the degree sequence of G. The graph union G ∪ H of two graphs G and H with disjoint vertex sets is another The union of s disjoint copies is denoted by sG. For terms and notations not defined here we follow the standard texts in graph theory as the famous book of West [19].
The first Zagreb index of a graph G is an old degree-based graph invariant introduced by Gutman and Trinajstić [8] In a recent paper about the general form of all degree-based topological indices of graphs [9], Gutman introduced two new invariants and invited researchers to investigate their mathematical properties and chemical meanings. He used the names "Sombor index" and "reduced Sombor index" for his new graphical invariants. The Sombor and reduced Sombor indices are defined as follows: We refer to [15,20], for more information on degree-based topological indices of graphs and their extremal problems.
Let Γ(s, n) denote the set of all non-increasing real sequences c = (c 1 , c 2 , . . . , c n ) such that n i=1 c i = s. Define a relation on Γ(s, n) as follows: For two non-increasing real sequences c = (c 1 , c 2 , . . . , c n ) and d = (d 1 , d 2 , . . . , d n ) in Γ(s, n), we write c d if and It is easy to see that (Γ(s, n), ) is a partially ordered set. The partial order is called the majorization and if c d then we say that c majorized by d. We refer the interested readers to consult the survey article [12] and the book [13], for more information on majorization theory and its applications in graph theory.
Suppose X ⊆ R n and a, b ∈ X are different points in X. The line segment ab is the set of all points λa + (1 − λ)b, where 0 < λ < 1. The set X is said to be convex, if for every point a, b ∈ X, ab ⊆ X. Let X ⊆ R n be convex. The function f : X −→ R is called a convex function, if for any a, b ∈ X and 0 < λ < 1, we have f (λa . If f is convex and we have strict inequality for all a = b, then we say the function is strictly convex. It is well-known that if I is an open interval and g : I −→ R is a real twice-differentiable function on I, then g is convex if and only if for each x ∈ I, g ′′ (x) ≥ 0. The function g is strictly convex on I, if g ′′ (x) > 0 for all x ∈ I.

Background Materials
In [9], Gutman proved that among all n-vertex graphs, the empty graph K n and the complete graph K n have the minimum and maximum Sombor indices, respectively. He also proved that if we restrict our attention to the n-vertex connected graph then the n-vertex path P n will attain the minimum Sombor index. He also proved in [10] that Cruz et al. [1] characterized the graphs extremal with respect to the Sombor index over the set of all chemical graphs, connected chemical graphs, chemical trees, and hexagonal systems. Cruz and Rada [2], studied the extremal values of Sombor index over the set of all unicyclic and also bicyclic graphs of a given order. In a recent work, Das et al. [3] obtained lower and upper bounds for the Sombor index of graphs based on some other graph parameters. Moreover, they obtained some relationships between Sombor index and the first and second Zagreb indices of graphs.
Deng et al. [5], investigated the chemical importance of the Sombor index and obtained the extremal values of the reduced Sombor index for chemical trees. Milovanović et al. [14] investigated the relationship between Sombor index and Albertson index which is an old irregularity measure for graphs, and in [16], Redzepović examined the predictive and discriminative potentials of Sombor and reduced Sombor indices of chemical graphs.
Wang et al. [18] investigated the relationships between the Sombor index and some degree based invariants, and obtained some Nordhaus-Gaddum type results. In [4], the authors presented some bounds on Sombor index of trees in terms of graph parameters and characterized the extremal graphs.
The following lemma [11] is useful in some of our results.
We denote by H n,ν the graph constructed from the star S n by adding ν edge(s) , between a fixed pendent vertex and ν other pendent vertices [17]. This conjecture will be proved in Section 3.

Main Results
The aim of this section is to prove Conjecture 2.2. It is also proved for the reduced Sombor index. The relationship between the Sombor index, reduced Sombor index and the first Zagreb index of graphs will also be investigated. Proof. Dimitrov and Ali [6], proved that  For a graph G, we define Proof. For any uv ∈ E(G), we have deg G (u) + deg G (v) ≤ m + 1. Note that the function g(x) = (x + 1) 2 is strictly convex on (−∞, ∞), and so for every a, b, a ≥ b ≥ 2, we have

Moreover, the above equality holds if and only if deg
Lemma 3.4. Let G be a graph with cyclomatic number ν (0 ≤ ν ≤ n − 2) and vertex set with equality if and only if G ∼ = H n,ν .
Theorem 3.8. Suppose G has maximum reduced Sombor index among all n-vertex graphs with cyclomatic number ν. If 0 ≤ ν ≤ n − 2, then G ∼ = H n,ν and SO red ( Proof. The proof follows from Lemma 3.6 and a similar argument as Theorem 3.5.
The following lemma is useful in finding some new lower bounds for the Sombor and reduced Sombor indices of graphs. .
Suppose x and y are two positive real numbers and x + y − 2 = s. Since The equalities hold if and only if G ∼ = P n or G ∼ = C n .
Proof. By definition of Sombor index, and by our discussion before the statement of this theorem, Now, by Lemma 3.9, The equality holds if and only if G ∼ = P n or G ∼ = C n . By definition of reduced Sombor Again by our discussion before the statement of this theorem, SO red (G) ≥ m 1,2 + 2m 1,3 + √ 2m 2,2 +