The Frank graph

Abstract

This paper introduces the Frank graph, a special weighted digraph with self-loops whose adjacency matrix corresponds to the Frank matrix, a special Max matrix. We derive bounds for the spectral radius and energy of the graph and illustrate these results with numerical examples. By investigating the spectral properties and extending the analysis of energy to weighted digraphs with self-loops, this study provides new insights and serves as a reference for future research on graphs with specialized adjacency matrices.