Laplacian Eigenvalues of Character Degree Graphs of Solvable Groups

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