  Given a finite group $G$, the character graph, denoted by $\Delta(G)$, for
its irreducible character degrees is a graph with vertex set $\rho(G)$ which is
the set of prime numbers that divide the irreducible character degrees of $G$,
and with $\{p,q\}$ being an edge if there exist a non-linear $\chi\in {\rm
Irr}(G)$ whose degree is divisible by $pq$. In this paper, we discuss the
influences of cut vertices and eigenvalues of $\Delta(G)$ on the group
structure of $G$. 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 $\Delta(G)$.
