Research Article

The Frank graph

Published in: Quaestiones Mathematicae
Volume 49 , issue 5 , 2026 , pages: 619–639

DOI: 10.2989/16073606.2025.2607410

Author(s): Efruz Özlem Mersin Aksaray University, Turkey , Ayşe Dilek Maden Selcuk University, Turkey , Mustafa Bahşi Aksaray University, Turkey

Keywords: 05C20 , 05C22 , 05C50 , Frank graph , Frank matrix , spectral radius , weighted digraph

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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.

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