The fundamental group of the clique graph

AbstractGiven a finite connected bipartite graph B=(X,Y) we consider the simplicial complexes of complete subgraphs of the square B2 of B and of its induced subgraphs B2[X] and B2[Y]. We prove that these three complexes have isomorphic fundamental groups. Among other applications, we conclude that the fundamental group of the complex of complete subgraphs of a graph G is isomorphic to that of the clique graph K(G), the line graph L(G) and the total graph T(G)

On the character degree graph of solvable groups

Let G be a finite solvable group, and let 06(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. A fundamental result by P. P. P\ue1lfy asserts that the complement 06(G) of the graph 06(G) does not contain any cycle of length 3. In this paper we generalize P\ue1lfy\u2019s result, showing that 06(G) does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of 06(G) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if n is the clique number of 06(G), then 06(G) has at most 2n vertices. This confirms a conjecture by Z. Akhlaghi and H. P. Tong-Viet, and provides some evidence for the famous \u3c1-\u3c3 conjecture by B. Huppert

Groups acting on quasi-median graphs. An introduction

Quasi-median graphs have been introduced by Mulder in 1980 as a generalisation of median graphs, known in geometric group theory to naturally coincide with the class of CAT(0) cube complexes. In his PhD thesis, the author showed that quasi-median graphs may be useful to study groups as well. In the present paper, we propose a gentle introduction to the theory of groups acting on quasi-median graphs.Comment: 16 pages. Comments are welcom