Some bounds on the Aα-index of connected graphs with fixed order and size

摘要

Let A(G) and D(G) be the adjacency matrix and the degree diagonal matrix of a graph G, respectively. For any real alpha is an element of [0, 1], Nikiforov [Appl Anal Discrete Math. 2017;11:81- 107] proposed the matrix A(alpha)(G) = alpha D(G) + (1 - alpha)A(G), whose largest eigenvalue is called the A(alpha)-index of G. In this contribution, we study this novel and interesting matrix. On the one hand, we show that in the set of connected graphs with fixed order and size, the graphs with the maximum A(alpha)-index are the nested split graphs (i.e. threshold graphs). On the other hand, we establish some upper and lower bounds on the A(alpha)-index of nested split graphs using eigenvector techniques. Finally, we present some computational results to compare these bounds.

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