Generalized Randić Estrada Indices of Graphs

Abstract

Let G be a simple undirected graph on n vertices. V. Nikiforov studied hybrids of AG and DG and defined the matrix AαG for every real α∈[0,1] as AαG=αDG+(1−α)AG. In this paper, we define the generalized Randić matrix for graph G, and we introduce and establish bounds for the Estrada index of this new matrix. Furthermore, we find the smallest value of α for which the generalized Randić matrix is positive semidefinite. Finally, we present the solution to the problem proposed by V. Nikiforov. The problem consists of the following: for a given simple undirected graph G, determine the smallest value of α for which AαG is positive semidefinite.

Keywords

• convex combination of matrices
• generalized Randić matrix
• Estrada index
• positive semidefinite matrix
• bipartite graph