Abstract
Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of A(G) and D(G) and defined the Aα-matrix for every real α∈[0,1] as: Aα(G)=αD(G)+(1−α)A(G). In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the Aα-matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established.
Highlights
Throughout the paper, we consider G an arbitrary connected graph with the edge set denoted by E (G) and its vertex set V(G) = {1, . . . , n} with cardinality m and n, respectively
The adjacency matrix A(G) of the graph G is a symmetric matrix of order n with entries aij, such that aij = 1 if ij ∈ E (G) and aij = 0 otherwise
Considering Theorem 3 and the results previously shown, we allow obtaining new lower bounds for the Estrada index of the Aα-matrix
Summary
We denote the eigenvalues of the Laplacian matrix by μ1 ≥ μ2 ≥ . Notice that we can obtain a lower bound for the Estrada index considering (3) and (4) by:. The Estrada index of the Aα-matrix of graph G is defined as n. In 1985, Anderson et al [3] obtained the following upper bound for the Laplacian matrix. As a consequence of the inequality (10), Lemma 3, and Theorem 1, we have the following result. The equality case on both inequalities is attained if and only if α = 0 and G is isomorphic to Kn. In this paper, new lower bounds for the Estrada index are established. Considering Theorem 3 and the results previously shown, we allow obtaining new lower bounds for the Estrada index of the Aα-matrix
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