首页|A relation between a vertex-degree-based topological index and its energy
A relation between a vertex-degree-based topological index and its energy
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NSTL
Elsevier
Let G be a simple graph with vertex set V and edge set E, and let di be the degree of the vertex v(i) is an element of & nbsp;V. If the vertices vi and v(j) are adjacent, we denote the respective edge by v(i)v(j) & nbsp;is an element of & nbsp; E. A vertex-degree-based topological index phi is defined as phi(G) = sigma(vivj is an element of E)& nbsp;phi(di, dj), where phi(i,j )is a function with the & nbsp;(& nbsp;)property phi(i,j) = phi(j,i). The general extended adjacency matrix A(phi) is defined as [A(phi)](ij )= phi(di,dj) if v(i)v(j)& nbsp;is an element of & nbsp;E, and 0 otherwise. The energy associated to phi of G is the sum of the absolute values of the eigenvalues of A(phi).& nbsp;In this paper we show that rho(G) epsilon(phi)(G) >=& nbsp;2(phi)(G) for all connected graphs G, where rho(G) is the spectral radius of G and phi(a,b) &NOTEQUexpressionL; 0 for all a, b & nbsp;is an element of & nbsp; N. We also characterize the graphs where equality holds. As a consequence, for any tree T with n vertices, E-phi(T)& nbsp;>=& nbsp;& nbsp;root 2./n - 1 phi(1,(n-1)), with equality holding if and only if T expressionpproximexpressiontely equexpressionl to & nbsp;S-n. (C)& nbsp;2021 Elsevier Inc. All rights reserved.& nbsp;
NSTL
Elsevier
