On \(\alpha\) – Incidence Energy and \(\alpha\)- Distance Energy of a Graph

Abstract

The \(\alpha\)-incidence energy of a graph is defined as the sum of \(a\)th powers of the signless Laplacian eigenvalues of the graph, where \(a\) is a real number such that \(\alpha \neq 0\) and \(\alpha \neq 1\). The \(\alpha\)-distance energy of a graph is defined as the sum of \(a\)th powers of the absolute values of the eigenvalues of the distance matrix of the graph, where \(\alpha\) is a real number such that \(\alpha \neq 0\). In this note, we present some bounds for the \(\alpha\)-incidence energy of a graph. We also present some bounds for the \(\alpha\)-distance energy of a tree.