Bounds of modified Sombor index, spectral radius and energy

Let G be a simple graph with edge set E(G). The modified Sombor index is defined as mS O(G) = ∑ uv∈E(G) 1 √ d2 u+d v , where du (resp. dv) denotes the degree of vertex u (resp. v). In this paper, we determine some bounds for the modified Sombor indices of graphs with given some parameters (e.g., maximum degree ∆, minimum degree δ, diameter d, girth g) and the Nordhaus-Gaddum-type results. We also obtain the relationship between modified Sombor index and some other indices. At last, we obtain some bounds for the modified spectral radius and energy.


Introduction
Recently, a novel topological index, named as Sombor index [13], was introduced by Gutman, which is defined as The maximum chemical trees with respect to Sombor indices were determined by Deng [10] and Cruz [6], independently. And then, present authors [22] characterized 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. Chemical applicability of Sombor indices was studied by Redžepović [34]. Milovanović et al. [32] determined relations between Sombor indices and other indices. Other results about Sombor index can be found in [1, 5, 9, 12-14, 18, 21, 23, 24, 35, 38]. Some other results about topological indices can see [30,31] and so on.
Kulli and Gutman also proposed the modified Sombor index [19], which is defined as In this paper, we determine some bounds for the modified Sombor indices of graphs with given some parameters (e.g., maximum degree ∆, minimum degree δ, diameter d, girth g) and the Nordhaus-Gaddum-type results. We also obtain the relationship between modified Sombor index and some other indices. At last, we obtain some bounds for modified Sombor spectral radius and energy. In this paper, all notations and terminologies used but not defined here can refer to Bondy and Murty [4]. [25,27]) and by the definition of modified Sombor index, we have Theorem 2.1. Let G be a simple graph with maximum degree ∆, minimum degree δ and Randic index R(G). Then

Bounds for the modified Sombor indices of graphs
with equality iff G is a regular graph.
, and it follows that m S O(G) ≤ √ 2 2 R(G), with equality iff d u = d v for every edge uv ∈ E(G), i.e., G is a regular graph.
On the other hand, we have m S O(G) = , with equality iff G is a regular graph.
. It is known that R(G) ≤ n 2 , with equality iff G is a regular graph (see [2,7,39]  Denote by χ = uv∈E(G) Theorem 2.6. Let G be a simple graph with m edges and minimum degree δ.
with left equality iff G is a regular graph, right equality iff G is a semiregular graph.
Proof. Since for every edge uv ∈ E(G), i.e., G is a semiregular graph (see [16]). On the other hand, we have It is obvious that 1 . By the definition of modified Sombor index, we have Theorem 2.7. Let G be a simple graph with m edges, maximum degree ∆ and minimum degree δ.
with equality iff G is a regular graph.
Corollary 2.8. Let G be a simple graph with n vertices, maximum degree ∆ and minimum degree δ.
with equality iff G is a regular graph.
Proof. It is obvious that nδ ≤ v∈V(G) Then by Theorem 2.7, we obtain the results as desired.
In the following, we discuss the relation between modified Sombor index and general Randic index Theorem 2.9. Let G be a simple graph with maximum degree ∆, minimum degree δ and general Randic index with equality iff G is a regular graph.
G is a regular graph. Similarly, we can obtain the corresponding upper bound.
In the following, we get the relation between modified Sombor index and Harmonic index. The Harmonic index [11] is defined as Theorem 2.10. Let G be a graph with n vertices and the Harmonic index H(G). Then with left equality iff G K n , and right equality iff G is a regular graph.
In the following, we obtain the bounds of modified Sombor index with given some parameters.
Corollary 2.11. Let G be a connected graph with n vertices and diameter D(G).
, with equality iff G K n (see [20]). Combining these with Theorem 2.10, we obtain the desired results. (m + 2n), with equality iff G C n .
Proof. Notice that H(G) ≤ m+2n 6 , with equality iff G P n or C n (see [36]). Combine with Theorem 2.10, we obtain the conclusion.
and (2) with equalities iff G C n , (3) and (4) with equalities iff G is a regular graph and g(G) = k. (1) and (2) with equalities iff G C n . (3) and (4) with equalities iff G is a regular graph and g(G) = k (see [40]). Combine with Theorem 2.10, we obtain the conclusion.
Corollary 2.14. Let G be a connected graph with n vertices and m edges. Denote ν 1 the largest eigenvalue of the sum-connectivity matrix S (G). Then with equality iff G K n .
Proof. Since H(G) ≥ n n−1 ν 2 1 , with equality iff G K n or K n (see [43]). Combine with Theorem 2.10, we obtain the conclusion.
A more precise result for Theorem 2.1 is given as follows.
Corollary 2.15. Let G be a connected graph with n vertices, m edges and Randic index R(G). Then with equality iff G is a 2m n -regular graph. Proof. Since H(G) ≤ R(G), with equality iff G is a 2m n -regular graph (see [39]). Combine with Theorem 2.10, we obtain the conclusion. Proof. Since H(G) ≤ 1 2 ABC(G) + R(G), with equality iff G P 2 (see [37]). Combine with Theorem 2.10, we obtain the conclusion.
Corollary 2.17. Let G be a connected graph. The minimum degree of G is at least k ≥ 2. Let X(G) be the sum-connectivity index of G. Then 2k−2 with equality iff G is a k-regular graph (see [41]). Combine with Theorem 2.10, we obtain the conclusion.
Corollary 2.18. Let G be a simple graph with minimum degree δ, maximum degree ∆ and geometricarithmetic index GA(G). Then

GA(G)
2∆ with left equality iff G K n , right equality iff G is a regular graph.
with equality iff G is a regular graph (see [28]). Combine with Theorem 2.10, we obtain the conclusion.
Corollary 2.19. Let G be a simple graph with minimum degree δ, maximum degree ∆ and geometricarithmetic index GA(G). Then with left equality iff G K n , right equality iff G is a regular graph.
, with equality iff G is a regular graph (see [29]). Combine with Theorem 2.10, we obtain the conclusion.  Proof. Since H(G) ≤ 1 2 ID(G)∆, with equality iff G is a regular graph (see [8]). Combine with Theorem 2.10, we obtain the conclusion. with equality iff G is a regular graph.
with left equality iff G K n , right equality iff G is a regular graph.
Proof. Since 2IS I(G) with equality iff G is a regular graph (see [26]). Combine with Theorem 2.10, we obtain the conclusion.
In the following, we obtain a Nordhaus-Gaddum-type result with respect to modified Sombor index. with equality iff G K n or G K n .
Proof. Since |E(G)| + |E(G)| = n 2 and . By the definition of modified Sombor index, we have

Bounds for the modified Sombor spectral radius and energy
Denote by λ 1 (G) ≥ λ 2 (G) ≥ · · · ≥ λ n (G) the eigenvalues of adjacent matrix A(G) of G. Let E A (G) = n i=1 |λ i | be the energy of G (see [15]). The modified Sombor matrix can be denoted by Denote by µ 1 (G) ≥ µ 2 (G) ≥ · · · ≥ µ n (G) the eigenvalues of modified Sombor matrix S (G) of G. Let E S (G) = n i=1 |λ i | be the modified Sombor energy of G. Theorem 3.1. Let G be a simple connected graph with n (n ≥ 3) vertices, maximum degree ∆ and minimum degree δ.
with equality iff G is a connected regular graph.
Proof. (1) Let X = (x 1 , x 2 , · · · , x n ) be a unit eigenvector of G corresponding to λ 1 . By Rayleigh theorem, then µ 1 ≥ X T S X X T X = X T S X = 2 (2) Let Y = (y 1 , y 2 , · · · , y n ) be a unit eigenvector of G corresponding to µ 1 . By Rayleigh theorem, 2δ , with equality iff G is a connected regular graph.
Corollary 3.2. Let G be a simple connected graph with n (n ≥ 3) vertices, maximum degree ∆ and minimum degree δ. Z 1 (G) is the first Zagreb index. Then with equality iff G is a connected regular graph.
Proof. Since Z 1 (G) n ≤ λ 1 ≤ ∆, with left equality iff G is a regular graph or semiregular graph, with right equality iff G is a regular graph (see [42]). Combine with Theorem 3.1, we obtain the conclusion. Corollary 3.3. Let G be a simple connected graph with n (n ≥ 3) vertices, m edges, maximum degree ∆ and minimum degree δ. Then with equality iff G is a connected regular graph.
Theorem 3.4. Let G be a simple graph with n vertices. Then Lemma 3.5. Let G be a simple graph with n vertices. The modified Sombor matrix of G is S (G) and it's modified Sombor eigenvalues µ 1 ≥ µ 2 ≥ · · · ≥ µ n . Then (2) tr( ); ).

Concluding remarks
In chemical graph theory, it is important to study the extremal structure of graph with respect to the topological indices, because it provides some useful information to determine the applicability range of the topological indices in QSPR and QSAR. In this paper, we obtain some bounds for modified Sombor indices of graphs with given some parameters and Nordhaus-Gaddum-type results. Then, we obtain some bounds for the modified Sombor spectral radius and energy. It is interesting to consider other properties of modified Sombor index. Therefore, we propose the following problems.