On cut vertices and eigenvalues of character graphs of solvable groups

Given a finite group G, the character graph, denoted by Δ(G), for its irreducible character degrees is a graph with vertex set ρ(G) which is the set of prime numbers that divide the irreducible character degrees of G, and with {p,q} being an edge if there exists a non-linear χ∈Irr(G) whose degree is divisible by pq. In this paper, on one hand, we proceed by discussing the graphical shape of Δ(G) when it has cut vertices or small number of eigenvalues, and on the other hand we give some results on the group structure of G with such Δ(G). Recently, Lewis and Meng proved the character graph of each solvable group has at most one cut vertex. Now, we determine the structure of character graphs of solvable groups with a cut vertex and diameter 3. Furthermore, we study solvable groups whose character graphs have at most two distinct eigenvalues. Moreover, we investigate the solvable groups whose character graphs are regular with three distinct eigenvalues. In addition, we give some lower bounds for the number of edges of Δ(G).