Abstract
Let G be a graph with set of vertices V(G)(|V(G)|=n) and edge set E(G). Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by SO(G) and is defined as SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of the vertex vi in G. In this paper we present some lower and upper bounds on the Sombor index of graph G in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs.
Highlights
Let G = (V, E) be a graph with vertex set V(G) = {v1, v2, . . . , vn} and edge set E = E(G), where |V(G)| = n and |E(G)| = m
For a subset W of V(G), let G − W be the subgraph of G obtained by deleting the vertices of W and the edges incident with them
If W = {vi} and E = {vivj}, the subgraphs G − W and G − E will be written as G − vi and G − vivj for short, respectively
Summary
For any two nonadjacent vertices vi and vj of a graph G, we let G + vivj be the graph obtained from G by adding the edge vivj. Many fundamental mathematical properties such as lower and upper bounds can be found in, e.g., [3,10,18–26] This topological index was motivated by the geometric interpretation of the degree radius of an edge vivj, which is the distance from the origin to the ordered pair (dG(vi), dG(vj)). Denote by W (n, ω) the set of connected graphs of order n with clique number ω. Let X (n, k) be the set of connected graphs of order n with chromatic number k. Recall that a short kite graph Kinω obtained by adding n − ω pendant vertices to the unique vertex of clique Kω; see Figure 2.
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