Abstract
Graph energy is an invariant that is derived from the spectrum of the adjacency matrix of a graph. Graph energy is actually the absolute sum of all the eigenvalues of the adjacency matrix of a graph i.e. E = ∑ i = 1 n λ i , and the Estrada index of a graph G is elaborated as EE G = ∑ i = 1 n e λ i , where, λ 1 , λ 2 , … , λ n are the eigenvalues of the adjacency matrix of a graph. In this paper, energy E G and Estrada index EE G of different molecular structures are obtained and also established inequalities among the exact and estimated values of energies and Estrada index of TUC 4 C 8 nanosheet and naphthalene.
Highlights
Chemical graph theory is a branch of mathematical chemistry and mathematical chemistry gives a natural way to understand the mathematical structures which act behind the existing chemical concepts to inaugurate the novel mathematical ideas and mathematical techniques in chemistry. e chemical graph theory is the intersection of graph theory and chemistry
One can pick the molecular structure of different compounds and transform them into a molecular graph and apply different graph invariants such as the energy of a graph. is graph invariant is much more popular in both mathematicians and chemists and was invented in 1978 by Ivan Gutman [1]
Mathematicians were not interested initially but when the 20th century was started, a dramatic change occurred and graph energy got great respect from the whole world when the number of countries such as Pakistan, Australia, Austria, Brazil, Canada, and many more produced numbers of papers in graph energy. e definition of graph energy is motivated by already existing results for the Huckel molecular orbital total π-electron energy [2, 3]. e energy of a graph defined here totally coincides with the total π-electron energy computed using Huckel molecular orbital theory
Summary
Chemical graph theory is a branch of mathematical chemistry and mathematical chemistry gives a natural way to understand the mathematical structures which act behind the existing chemical concepts to inaugurate the novel mathematical ideas and mathematical techniques in chemistry. e chemical graph theory is the intersection of graph theory and chemistry. E definition of graph energy is motivated by already existing results for the Huckel molecular orbital total π-electron energy [2, 3]. E energy of a graph defined here totally coincides with the total π-electron energy computed using Huckel molecular orbital theory. E application of the Estrada index is to measure the folding degree of long-chain proteins [5,6,7,8]. Journal of Chemistry e motivation of the paper was to find the energy and Estrada index of nanostructures. We find the energy of a graph by finding the eigenvalues of its adjacency matrix. Λn and the set of these eigenvalues is called a spectrum of the graph and since the adjacency matrix is symmetric so all the eigenvalues are real Let G be a simple undirected graph in which V denotes the vertex set containing v1, v2, . . . , vn as atoms and E denotes the edge set e1, e2, . . . , en as bonds. e adjacency matrix A(G) of a graph G is a square matrix that can be obtained by the adjacent and nonadjacent vertices in such a way if (i, j)entry of a matrix is 1 whenever the vertices vi and vj are adjacent, otherwise 0. e eigenvalues of the adjacency matrix of order n are λ1, λ2, . . . , λn and the set of these eigenvalues is called a spectrum of the graph and since the adjacency matrix is symmetric so all the eigenvalues are real
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