Abstract
For a finite group G, let R(G) be the solvable radical of G. The character-graph of G is a graph whose vertices are the primes which divide the degrees of some irreducible complex characters of G and two distinct primes p and q are joined by an edge if the product pq divides some character degree of G. In this paper we prove that, if has no subgraph isomorphic to and it’s complement is non-bipartite, then is an almost simple group with socle isomorphic to where is a prime power. Also we study the structure of all planar graphs that occur as the character-graph of a finite group G.
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