Nordhaus–Gaddum-Type Relations for Arithmetic-Geometric Spectral Radius and Energy.

Spectral graph theory plays an important role in engineering. Let G be a simple graph of order n with vertex set V = v 1 , v 2 , ... , v n . For v i ∈ V , the degree of the vertex v i , denoted by d i , is the number of the vertices adjacent to v i . The arithmetic-geometric adjacency matrix A a g G of G is defined as the n × n matrix whose i , j entry is equal to d i + d j / 2 d i d j if the vertices v i and v j are adjacent and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of G are the spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus–Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs. [ABSTRACT FROM AUTHOR]