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In the last decade, several scholars proposed an unifying approach to study the spectral theories of the adjacency, Laplacian and signless Laplacian of graphs. The most general graph matrix is the  universal adjacency matrixU  =  αA  +  βD  +  γJ  +  δI , where  A ,  D ,  J , and  I  are the adjacency matrix of  G , the degree matrix of  G , the all-ones matrix, the identity matrix, respectively. Here, we consider, with  β  ≥ 0, and we study the graphs belonging to some given class Γ maximizing the corresponding spectral radius. In particular, we consider connected graphs with prescribed  c -cyclic degree sequence,  c  ∈ {0, 1, 2}, and the multicone graphs defined over them, where the multicone graph is the join of a clique with a given graph. The aim of this paper is to provide the best possible generalization of results to the spectral radius of(and the graph matrices related to it) of several well-known results for multicone graphs over connected graphs with prescribed  c -cyclic degree sequence, where  c  ∈ {0, 1, 2}.

The general spectral radii of (multicone-)graphs with prescribed degree sequence