Spectral radius of power graphs on certain finite groups

The power graph of a group $G$ is a graph with vertex set $G$ and two distinct vertices are adjacent if and only if one is an integral power of the other. In this paper we find both upper and lower bounds for the spectral radius of power graph of cyclic group $C_n$ and characterize the graphs for which these bounds are extremal. Further we compute spectra of power graphs of dihedral group $D_{2n}$ and dicyclic group $Q_{4n}$ partially and give bounds for the spectral radii of these graphs.     

The smallest surface that contains all signed graphs on K4,n

Finding the smallest number of crosscaps that suffice to orientation-embed every edge signature of the complete bipartite graph $K_{m,n}$ is an open problem. In this paper that number for the complete bipartite graph $K_{4,n}$, $n\ge 4$, is determined by using diamond products of signed graphs. The number is $2?\frac{n?1}{2}?+1$, which is attained by $K_{4,n}$ with exactly 1 negative edge, except that when $n=4$, the number is 4, which is attained by $K_{4,4}$ with exactly 4 independent negative edges.