On weighted unicyclic graphs with prescribed degree sequence and weight set with maximal spectral radius

Abstract

Let π = (d1,d2, ....,dn) be a unicyclic graphic degree sequence such that d1 > d2 ≥ .... ≥ dn and W = {w1, w2, ... wn} be a positive weight set such that w1 ≥ w2 ≥ .... > wn ≥ 0. In this paper, It is shown that the extremal graphs with the largest spectral radius among all weighted unicyclic graphs with a given degree sequence n and a positive weight set IV have the same structural property. Moreover, such extremal graphs are characterized when w1 = w2 = .... = wn or w2≥ w3d1d4.