Two spectral extremal results for graphs with given order and rank

TL;DR

This paper investigates the extremal spectral radius of graphs with fixed order and rank, identifying graphs that maximize or minimize spectral radius for given parameters.

Contribution

It generalizes previous results by determining extremal graphs with maximum spectral radius for any fixed order and rank, and minimum spectral radius for rank 5.

Findings

Identified extremal graphs with maximum spectral radius for given order and rank.

Determined the extremal graph with minimum spectral radius for graphs of order n and rank 5.

Abstract

The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum or minimum spectral radius over certain families of graphs. Monsalve and Rada [Extremal spectral radius of graphs with rank 4, Linear Algebra Appl. 609 (2021) 1-11] obtained the extremal graphs with maximum and minimum spectral radii among all graphs with order n and rank 4. In this paper, we first determine the extremal graph which attains the maximum spectral radius among all graphs with any given order n and rank r, and further determine the extremal graph which attains the minimum spectral radius among all graphs with order n and rank 5.