Laplacian Eigen values of character degree graphs of solvable groups

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.