Nordhaus-Gaddum-type relations for arithmetic-geometric spectral radius and energy
Summary: Spectral graph theory plays an important role in engineering. Let \(G\) be a simple graph of order \(n\) with vertex set \(V=\left\{ v_1, v_2, \ldots, v_n\right\}\). For \(v_i\in V \), 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 \(A_{a g}\left( G\right)\) of \(G\) is defined as the \(n\times n\) matrix whose \(\left( i, j\right)\) entry is equal to \(\left( \left( d_i + d_j\right)/2 \sqrt{ d_i d_j}\right)\) 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.
scientific article; zbMATH DE number 7226148