Some results on the Sombor indices of graphs

This paper is concerned with three recently introduced degree-based graph invariants; namely, the Sombor index, the reduced Sombor index and the average Sombor index. The first aim of the present paper is to give some results that may be helpful in proving a recently proposed conjecture concerning the Sombor index. Establishing inequalities related to the aforementioned three graph invariants is the second aim of this paper.


Introduction
The study of the mathematical aspects of the degree-based graph invariants (also known as topological indices) is considered to be one of the very active research areas within the field of chemical graph theory [17]. Recently, the mathematical chemist Ivan Gutman [18], one of the pioneers of chemical graph theory, proposed a geometric approach to interpret degree-based graph invariants and based on this approach, he devised three new graph invariants; namely the Sombor index, the reduced Sombor index and the average Sombor index. The Sombor index, being the simplest one among the aforementioned three invariants, has attracted a significant attention from researchers within a very short time [3, 7-9, 14, 15, 19, 23, 26-31, 34, 35, 39, 41, 42].
The first aim of this paper to give some results that may be helpful in proving a conjecture concerning the Sombor index posed in the reference [35]. In order to state this conjecture, we need some definitions first. An acyclic graph is the graph containing no cycle. For a graph G, its cyclomatic number ν(G) (or simply ν) is the least number of edges whose deletion makes the graph G as acyclic. A ν-cyclic graph is the one having the cyclomatic number ν. A pendent vertex of a graph is a vertex of degree 1. For ν ≥ 1, denote by H n,ν the graph deduced from the star graph of order n by adding ν edge(s) between a fixed pendent vertex and ν other pendent vertices. Conjecture 1.1. [35] For the fixed integers n and ν with 6 ≤ ν ≤ n − 2, H n,ν is the only graph attaining the maximum Sombor index in the class of all ν-cyclic connected graphs of order n.
Establishing inequalities related to the Sombor index, the reduced Sombor index and the average Sombor index is another aim of this paper.

Preliminaries
Let G be a graph. Denote by E(G) and V (G) the edge set and vertex set, respectively, of G. Denote by i ∼ j the edge connecting the vertices v i , v j ∈ V (G). For a vertex v i ∈ V (G), its degree is denoted by d vi (G) (or simply by d i (G)). A regular graph is the one in which all of its vertices have the same degree. For an edge e ∈ E(G), its degree is the number of edges adjacent to e. By an edge-regular graph, we mean a graph in which all of its edges have the same degree. A graph of order n is also known as an n-vertex graph. Denote by G − v i and G − v i v j the graphs obtained from G by removing the vertex v i and the edge v i v j , respectively. The n-vertex complete graph and the n-vertex star graph are denoted as K n and K 1,n−1 , respectively. From the notations E(G), V (G), ν(G) and d i (G), we remove "(G)" whenever the graph under consideration is clear. The graph-theoretical notation and terminology used in this paper but not defined here, may be found in some standard graph-theoretical books, like [4,6,10].
If V (G) = {v 1 , v 2 , . . . , v n } and |E(G)| = m then the Sombor index, the average Sombor index and the reduced Sombor index of the graph G are defined as respectively. Most of the degree-based graph invariants can be written [22,38] as: where f is a symmetric non-negative real-valued function of d i and d j . The graph invariants having the form (1) are known as the bond incident degree indices [36], BID indices in short [2]. Those choices of the function f are given in Table 1 that correspond to the graph invariants used in the next sections.
forgotten topological index [16]  The p-Sombor index of a graph G is denoted by SO p (G) and is defined [35] as the sum of the quantities (d p i + d p j ) 1/p over all edges i ∼ j of G, where p is not equal to 0. The first lemma (Lemma 3.1) of this section gives an upper bound on a generalized variant SO p,q of the p-Sombor index: where q is a real number provided that ϕ p,q (i ∼ j) is a real number for every edge i ∼ j of G. The name (p, q)-Sombor index may be associated with the graph invariant SO p,q . Lemma 3.1. If G is a graph of size m ≥ 1 then for any real number q, it holds that with equality if and only if there exist a fixed real number t such that (d i + q) 2 + (d j + q) 2 = t for every edge i ∼ j of G, where F (G) and M 1 (G) are the forgotten topological index and first Zagreb index of G, respectively; see Table 1.
Proof. From Cauchy-Bunyakovsky-Schwarz's inequality, it follows that Note that the equality sign in (2) holds if and only if there exist a fixed real number t such that Next, the bound on the invariant SO 2,q given in Lemma 3.1 is used to derive another bound on SO 2,q in terms of the parameters m and q only (see Lemma 3.4); however, to proceed, bounds on the forgotten topological index F and first Zagreb index M 1 in terms of m are required first.
with equality if and only if the star S m+1 is a component of G.
Proof. We fix n and use induction on m. For m = 0, 1, the lemma is obviously true; thus, the induction starts. Suppose that G is an n-vertex graph of size k such that 0 ≤ k ≤ n − 1 and k ≥ 2. Take an edge i ∼ j. Without loss of generality, where the equality sign in (3) holds if and only if d i + d j = k + 1. (It needs to be mentioned here that, throughout this which forces that the right hand side of (3) is maximum if and only if d j = 1. Thus, (3) gives with equality if and only if d i = k and d j = 1. Also, by inductive hypothesis, it holds that with equality if and only if the star S k is a component of G − v i v j . Thus, from (4) and (5), it follows that F (G) ≤ k(k 2 + 1) with equality if and only if the star S k+1 is a component of G. This completes the induction and hence the proof.

Lemma 3.3. [40]
For any n-vertex graph G of size m with 0 ≤ m ≤ n − 1, it holds that with equality if and only if The next result follows directly from Lemmas 3.1, 3.2 and 3.3.

Lemma 3.4.
For any n-vertex graph G of size m with 0 ≤ m ≤ n − 1 and for any non-negative real number q, it holds that with equality if and only if the star S m+1 is a component of G.
The next two results were proven in [35].
Lemma 3.5. [35] If ν and n are fixed integers such that 0 ≤ ν ≤ n − 2 then the graph attaining the maximum Sombor index in the class of all connected ν-cyclic graphs of order n has the maximum degree n − 1.
Lemma 3.6. [35] Let ν and n be fixed integers such that 2 ≤ ν ≤ n − 2. Let G be a graph with the maximum value of the Sombor index in the class of all connected ν-cyclic graphs of order n.
For a real number α, define the graph invariant SO V,α as follows Towards the Proof of Conjecture 1.1. Let G be a ν-cyclic connected graph of order n with the vertex set is not adjacent to any non-pendent vertex of G then from either Lemma 3.5 or Lemma 3.6, respectively, it follows that G does not have the maximum value of SO in the class of all ν-cyclic connected graphs of order n. In what follows, assume that d 1 = n − 1 and that v 2 is adjacent to all non-pendent vertices of G. Note that the graph G − v 1 has exactly one connected non-trivial component C and the subgraph induced by V (C) (the vertex set of C) has a vertex of degree |V (C)| − 1, and that G − v 1 has the size m = ν, where 6 ≤ m ≤ n − 2 = |V (G − v 1 )| − 1. Thus, from Lemma 3.4, it follows that where the equality sign in (7) holds if and only if the star S ν+1 is a component of G − v 1 . We believe that the next result concerning the invariant SO V,α is true. However, at the present moment, we do not have its proof; thus, we state it as a conjecture (if one proves this conjecture then from (7), the proof of Conjecture 1.1 follows directly).
with equality if and only if the star S m+1 is a component of G.

Some relations between Sombor indices and other degree-based graph invariants
Before establishing the main results of this section, we first recall an inequality for the real number sequences reported in [33].
In the next theorem we determine a relationship between SO(G) and M 1 (G) and ISI(G).

Theorem 4.1. Let G be a connected graph. Then
Equality holds if and only if G is an edge-regular graph.
Proof. From the definitions of M 1 (G) and ISI(G) we have that On the other hand, for r = 1, x i := d 2 i + d 2 j , a i := d i + d j , with summation performed over all adjacent vertices v i and v j in G, the inequality (8) becomes , that is From the above and inequality (10) we arrive at (9).

Equality in (11) holds if and only if
√ d 2 i +d 2 j di+dj is constant for any pair of adjcent vertices v i and v j in G. Suppose that v j and v k are adjacent to vertex v i . Then From the above identity it follows that equality in (9) holds if and only if G is an edge-regular graph.
Equality holds if and only if G is an edge-regular graph.
Proof. The inequality (12) is obtained from (9) and which was proven in [12] Corollary 4.2. Let G be a connected graph with n ≥ 2 vertices and m edges. Then Equality holds if and only if G ∼ = K n or G ∼ = K 1,n−1 .
Proof. The inequality (13) is obtained from (12) and which was proven in [5].

Corollary 4.3. Let T be a tree with n ≥ 2 vertices. Then
Equality holds if and only if T ∼ = K 1,n−1 .
The inequality (14) was proven in [18]. Proofs of the following theorems are analogous to that of Theorem 4.1, thus omitted.
Equality holds if and only if G is an edge-regular graph.

Equality holds if and only if G is an edge-regular graph.
The next theorem reveals a connection between Sombor index and indices F (G), M 2 (G), AG(G) and GA(G).
Theorem 4.5. Let G be a connected graph. Then Equality holds if and only if G is regular.
Proof. The following identity holds By the arithmetic-geometric mean inequality (see e.g. [32]) we have that Combining (16) and (17) gives On the other hand, for r = 1, with summation performed over all adjacent vertices v i and v j in G, the inequality (8) transforms into Now, from the above and (18) we arrive at (15). Equality in (17) holds if and only if d i = d j for any pair of adjacent vertices v i and v j in G, which implies that equality in (15) holds if and only if G is regular.

Corollary 4.4. Let G be a connected graph. Then
Equality holds if and only if G is regular.
Proof. By the AM-GM inequality we have that The inequality (20) is obtained from the above and (15).

Equality holds if and only if G is regular.
Proof. The following is valid From the above and inequality (20) we obtain the required result.
The following inequality was proven in [24] for the real number sequences.
In the next theorem we determine a relationship between SO(G) and ID(G), F (G) and M 2 (G). Theorem 4.6. Let G be a connected graph. Then Equality holds if and only if G is an edge-regular graph.
Proof. For r = 2, p i := d 2 i + d 2 j , a i := 1 didj , with summation performed over all pairs of adjacent vertices v i and v j in G, the inequality (21) becomes On the other hand, for r = 1, x i := d 2 i + d 2 j , a i := d i d j , with summation performed over all pairs of adjacent vertices v i and v j in G, the inequality (8) becomes Now, from (23) and (24) we arrive at (22). Equality in (23) holds if and only if d i d j is constant for any pairs of adjacent vertices v i and v j in G. Suppose that vertices v j and v k are adjacent to v i . In that case, we have that This means that equality in (24) holds if and only if G is an edge-regular graph, which means that equality in (22) holds if and only if G is an edge-regular graph.
One can easily verify that from (24) the inequality (which was proven in [31]) follows. Corollary 4.6. Let G be a connected graph. Then Equality holds if and only if G is regular.
Theorem 4.7. Let G be a connected graph. Then Equality holds if and only if G is an edge-regular graph.
Proof. The following identities are valid and Taking r = 1, x i := |d i − d j |, and a i := d 2 i + d 2 j in inequality (8) with summation performed over all pairs of adjacent vertices v i and v j in G, we obtain Similarly, taking r = 1, x i := d i + d j , and a i := d 2 i + d 2 j in inequality (8) with summation performed over all pairs of adjacent vertices v i and v j in G,we obtain From the above inequalities we obtain the assertion of the Theorem 4.7.
Equality holds if and only if G is regular.
The inequality (26) was proven in [15,31] (see also [19]). By a similar arguments, the following results can be proven. Equality holds if and only if G is an edge-regular graph. Equality holds if and only if G is an edge-regular graph.
From Theorems 4.8 and 4.9 we have the following corollaries.
Equality holds if and only if G is regular or each of its components is regular.
Equality holds if and only if G is regular.