K₄-free character graphs with seven vertices

Abstract

For a finite group G, let ∆(G) denote the character graph built on the set of degrees of the irreducible complex characters of G. In this paper, we determine the structure of all finite groups G with K 4free character graph ∆(G) having seven vertices. We also obtain a classification of all K 4 -free graphs with seven vertices which can occur as character graphs of some finite groups.

n-exact Character Graphs

“…In this section, we wish to prove our main results. When G is a finite group with 4-exact character graph, then using [4], [7] and Lemma 2.11 we have nothing to prove. Hence in the sequel, we assume that G is a finite group such that for some integer n 5, ∆(G) is n-exact.…”

“…Lemma 2.9. [7] Let G be a finite group, R(G) < M G, S := M/R(G) be isomorphic to PSL 2 (q), where for some prime p and positive integer f 1,…”

Character graphs with diameter three

“…Lemma 3.5. [4] Suppose N is a Frobenius group whose kernel is an elementary abelian p-group. Then one of the following cases holds: a) θ is extendible to I and cd(G|θ) = {θ(1)[G : I], θ(1)b}, for some positive integer b divisible by (q 2 − 1)/(2, q − 1).…”

Disconnected Character graphs and odd Dominating sets

“…Lemma 2.4. [3] Let G be a finite group, R(G) < M G, S := M/R(G) be isomorphic to PSL 2 (q), where for some prime p and positive integer f…”