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.
2020 MATHEMATICS SUBJECT CLASSIFICATION: