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

Abstract EN

Spectral graph theory plays an important role in engineering.

Let G be a simple graph of order n with vertex set V=v1,v2,…,vn.

For vi∈V, the degree of the vertex vi, denoted by di, is the number of the vertices adjacent to vi.

The arithmetic-geometric adjacency matrix AagG of G is defined as the n×n matrix whose i,j entry is equal to di+dj/2didj if the vertices vi and vj 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.