Planar character-graphs

Abstract

For a finite group G, let R(G) be the solvable radical of G. The character-graph (Formula presented.) 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 (Formula presented.) has no subgraph isomorphic to (Formula presented.) and it’s complement is non-bipartite, then (Formula presented.) is an almost simple group with socle isomorphic to (Formula presented.) where (Formula presented.) is a prime power. Also we study the structure of all planar graphs that occur as the character-graph (Formula presented.) of a finite group G. © 2021 Taylor & Francis Group, LLC.