We characterize the finite graph algebras which are dualizable. Indeed, a finite graph algebra is dualizable if and only if each connected component of the underlying graph is either complete or bipartite complete (or a single point).

Let G be a finite insoluble group with soluble radical R(G). In this paper we investigate the graph of G, which is natural generalisation widely studied commuting graph. Here vertices are elements in G∖R(G), x adjacent to y if they generate subgroup G. Our main result states that always connected and its diameter, denoted δS(G), at most 5. More precisely, show δS(G)⩽3 not almost simple obtain s...

We prove a decomposition result for locally finite graphs which can be used to extend results on edge-connectivity from finite to infinite graphs. It implies that every 4k-edge-connected graph G contains an immersion of some finite 2k-edge-connected Eulerian graph containing any prescribed vertex set (while planar graphs show that G need not contain a subdivision of a simple finite graph of lar...