A finite group G is called Cayley integral if all undirected Cayley graphs over G are integral, i.e., all eigenvalues of the graphs are integers. The Cayley integral groups have been determined by Kloster and Sander in the abelian case, and by Abdollahi and Jazaeri, and independently by Ahmady, Bell and Mohar in the nonabelian case. In this paper we… (More)

For a finite group G and subset S of G, the Haar graph H(G,S) is a bipartite regular graph, defined as a regular G-cover of a dipole with |S| parallel arcs labelled by elements of S. If G is an abelian group, then H(G,S) is well-known to be a Cayley graph; however, there are examples of non-abelian groups G and subsets S when this is not the case. In this… (More)

In a recent paper (arXiv:1505.01475 ) Estélyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the… (More)

A Clar set of a benzenoid graph B is a maximum set of independent alternating hexagons over all perfect matchings of B. The Clar number of B, denoted by Cl(B), is the number of hexagons in a Clar set for B. In this paper, we first prove some results on the independence number of subcubic trees to study the Clar number of catacondensed benzenoid graphs. As… (More)