Laplacian Eigen values of character degree graphs of solvable groups

TL;DR

This paper investigates the Laplacian and distance Laplacian eigenvalues of character degree graphs of finite solvable groups, focusing on regular graphs, super graphs, and graphs with diameter two.

Contribution

It provides new results on the spectral properties of character degree graphs, including Laplacian eigenvalues for specific classes of these graphs.

Findings

Eigenvalues of regular character degree graphs are characterized.

Spectral properties of super graphs of these graphs are analyzed.

Character degree graphs with diameter two have a specific block structure.

Abstract

Let $G$ be a finite solvable group, let $Irr(G)$ be the set of all complex irreducible characters of $G$ and let $cd(G)$ be the set of all degrees of characters in $Irr(G).$ Let $\rho(G)$ be the set of primes that divide degrees in $cd(G).$ The character degree graph $\Delta(G)$ of $G$ is the simple undirected graph with vertex set $\rho(G)$ and in which two distinct vertices $p$ and $q$ are adjacent if there exists a character degree $r \in cd(G)$ such that $r$ is divisible by the product $pq.$ In this paper, we obtain Laplacian eigen values and distance Laplacian eigen values of regular character degree graph, super graphs of regular character degree graph and character degree graph with diameter $2$ has two blocks.