On spectral radius of generalized arithmetic–geometric matrix of connected graphs and its applications

Abstract

The generalized arithmetic–geometric matrix of a simple connected graph G=(V,E) is defined to be the |V|×|V| matrix whose ij-entry is diα+djα2diαdjα if vivj∈E, and 0 otherwise, where α is an arbitrary real number and di is the degree of vi∈V. This matrix is a general form of the arithmetic–geometric matrix (α=1) and the extended adjacency matrix (α=2), both of which have been well studied in spectral graph theory and chemical graph theory. In this paper, we focus on the spectral radius ρagα of the generalized arithmetic–geometric matrix of connected graphs. Some chemical applications of ρagα are explored, and some extremal results on ρagα are obtained; in particular, the connected graphs (including trees) having the maximum and minimum ρagα are determined.