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Abstract
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Let H be a finite group. The set of prime numbers that divide a degree of
the irreducible characters of H is denoted by ρ(H). The character graph ∆(H) has been
defined as a graph with the set of vertices ρ(H), and there exists an edge fr; sg in ∆(H)
if there is an irreducible character of H such that its degree is divisible by rs. Here, we
present several properties of ∆(H) from a graph theoretical perspective, including the
number of distinct eigenvalues and cut vertices. These properties provide insight into the
shape of ∆(H). Additionally, we provide bounds on the size and the matching number
of ∆(H).
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