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Abstract
Suppose G is a simple graph with edge set E G . The Randić index R G is defined as R G = ∑ u v ∈ E G 1 / deg G u deg G v , where deg G u and deg G v denote the vertex degrees of u and v in G , respectively. In this paper, the first and second maximum of Randić index among all n − vertex c − cyclic graphs was computed. As a consequence, it is proved that the Randić index attains its maximum and second maximum on two classes of chemical graphs. Finally, we will present new lower and upper bounds for the Randić index of connected chemical graphs.
Highlights
Materials is section aims to briefly review the literature on ordering graphs concerning the Randicindex
A chemical graph is a graph in which all vertices have degrees less than or equal to 4 [1]. e reason for this name is from quantum chemistry in which it is convenient to model a molecule M in such a way that vertices are used to denote atoms and edges are for chemical bonds
Caporossi et al [6] proved that among all 1-cyclic graphs of order n, the cycle Cn attains the maximum value, and the unicyclic graphs obtained by attaching a pendant path to a vertex of a cycle attain the second maximum Randicindex. ese are the starting point of the following problem
Summary
E reason for this name is from quantum chemistry in which it is convenient to model a molecule M in such a way that vertices are used to denote atoms and edges are for chemical bonds. E number of edges connecting a vertex of degree i with a vertex of degree j in G is denoted by mi,j(G). A connected n− vertex graph G is called to be c-cyclic if it has n + c − 1 edges and the number c c(G) is said to be the cyclomatic number of G. is topological index was proposed by Milan Randic [2] under the name “branching index.” e Randicindex is suitable for measuring the extent of branching of the carbonatom skeleton of saturated hydrocarbons. We encourage the interested readers to consult the books [3,4] for more information on this topic
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