Relating graph energy and Sombor index∗

The energy of a graph (ε) is the sum of absolute values of its eigenvalues, thus it is a graph-spectrum-based quantity. The Sombor index (SO) is a recently conceived vertex-degree-based topological index. We establish various relations between ε and SO, among which are lower and upper bounds. These relations improve and extend earlier results communicated in the paper [A. Ülker, A. Gürsoy, N. K. Gürsoy, MATCH Commun. Math. Comput. Chem. 87 (2022) 51–58].


Introduction
In this paper we are concerned with simple graphs, assumed to be connected. Let G be such a graph, with n vertices, m edges, vertex set V (G), and edge set E(G). The edge connecting the vertices u and v will be denoted by uv. The degree (= number of first neighbors) of the vertex u ∈ V (G) is denoted by d u . In addition, min u∈V (H) d u = δ and max u∈V (H) d u = ∆.
Let A(G) be the (0, 1)-adjacency matrix of the graph G, and let λ 1 , λ 2 , . . . , λ n be its eigenvalues, forming the spectrum of G [3]. The energy of the graph G is defined as This spectrum-based graph invariant has been extensively studied, both in mathematics [6][7][8] and in theoretical chemistry [5].
There are nowadays some 50 different bond incident degree (BID) graph invariants of the form where F is a suitably chosen function with the property F (x, y) = F (y, x). Among them, a very recently introduced such invariant is the Sombor index, for which F (x, y) = x 2 + y 2 . Thus, the Sombor index is defined as [4], At the first glance, there hardly could be expected that the spectrum-based graph energy ε and the vertex-degree-based indices BID(G) are anyhow related. Yet, the first such relation was discovered by Arizmendi and Arizmendi [1], who showed that ε(G) ≤ 2 R(G), where R(G) is the Randić index, F (x, y) = 1/ √ x y. This was followed by Yan et al. [11], who proved that ε(G) ≥ (2/∆) R(G). These results motivated three of the present authors to seek for analogous connections between ε and SO [10]. They obtained lower and upper bounds between ε and SO (for details see below). In the present paper we improve and extend these bounds, and offer a few more relations of this kind.

Energy of a vertex and its properties
In what follows we will much exploit the concept of energy of a vertex, invented by Arizmendi et al. in 2018 [2]. In this section we repeat the basic facts of their theory. Let the vertices of the graph G be labeled by v 1 , v 2 , . . . , v n . Then the energy of the vertex v i is defined as [2] ε( Equality holds if and only if v is the central vertex of K 1,n−1 . Proof. K 1,n−1 has m = n − 1 edges, whereas SO(K 1,n−1 ) = m √ m 2 + 1. The degree of the central vertex is equal to m, which by Theorem 2.1 is equal to ε(v) 2 .

Theorem 2.2. [2]
Let G be a graph with m ≥ 1 and maximum vertex degree ∆. Then for any Equality holds if and only if G ∼ = K a,a for a ≥ 1.

First inequality between graph energy and Sombor index
In [10], the following result was obtained: Proposition 3.1. [10] Let G be a connected graph with minimum vertex degree δ ≥ 2. Then ε(G) ≤ SO(G).
We now strengthen this result as follows: Proof. If n = 2, then G ∼ = K 2 . It is easy to compute that ε(K 2 ) = 2 and SO(K 2 ) = √ 2. Suppose now that n ≥ 3. Then the graph G cannot have an edge uv such that d u = d v = 1. Then the minimal value of the term d 2 Equality in (1) holds if and only if G ∼ = K 1,2 . The McClelland inequality is ε(G) ≤ √ 2mn [8,9]. For connected graphs, equality holds if and only if G ∼ = K 2 . Consider first the case that G is a tree. Then, n = m + 1, and √ 2mn = 2m(m + 1). It is now easy to show that 2m(m + 1) < √ 5 m holds for all m ≥ 1. If G is a connected cycle-containing graph, then n ≥ m, and Thus, in all cases, Bearing in mind the inequalities (1) and (2), we have We now improve Theorem 3.1, i.e., the inequality ε(G) 2 < SO(G) 2 .
Theorem 3.2. Let G be a connected graph with n ≥ 3 vertices. If δ ≥ 2, then Proof. Directly from the definition of Sombor index, he have If δ = 1, since d u = d v cannot happen, the left-hand inequality can be improved by Bearing in mind Theorem 2.1, Theorem 3.2 follows now by substituting (3) back into (4).

Second inequality between graph energy and Sombor index
In [10], the following result was obtained: Proposition 4.1. [10] Let G be a connected graph with maximum vertex degree ∆. Then ∆ 3 ε(G) ≥ SO(G).
We now establish a stronger upper bound for the Sombor index.
Equality holds if and only if G ∼ = K a,a for a ≥ 1.
Proof. By Theorem 2.2 we get with equality if and only if G ∼ = K a,a .
The term d 2 u + d 2 v is maximal if d u = d v = ∆, equal to √ 2 ∆. Then from the definition of Sombor index, it follows that SO is maximal if all vertex degrees are equal to ∆, i.e., if G is a regular graph. Thus with equality if and only if G is a regular graph. Substituting (6) back into (5), we get which directly leads to Theorem 4.1.