On the largest eigenvalue of some bidegreed graphs

Abstract

A simple connected non-regular graph is said to be -bidegreed, or biregular, if the vertices have degree from the set , with . We consider two classes of -bidegreed graphs denoted by and . A graph belongs to if: a) it is obtained from disjoint paths , , by identifying the vertices and the vertices , and the graph so obtained has n vertices; b) for each of the n vertices, pendant vertices are added, so that any vertex from any has degree . The class , , is similarly obtained by identifying all the vertices and from the ’s, into a single vertex. In this paper, we show that for any graph in or , the spectral radius of the adjacency matrix increases whenever the difference between the lengths of any two ’s increases. We also compute some bounds for the spectral radius when the lengths of the ’s tend to infinity. Finally, we discuss about bicyclic -bidegreed graphs with n degree vertices minimizing the spectral radius. We prove that in most cases such graphs do not belong to .