Some spectral inequalities for connected bipartite graphs with maximum [formula omitted]-index

Abstract

Let G be a graph, whose adjacency matrix and degree diagonal matrix are denoted by A(G) and D(G), respectively. In 2017 Nikiforov (2017) merged the A- and Q-spectral theories to proposed the Aα matrix: Aα(G)=αD(G)+(1−α)A(G), α∈[0,1]. Its largest eigenvalue is called the Aα-index of G. In this paper, we focus on specifying the property of the Aα matrix. On the one hand, we show that among the set of connected bipartite graphs of fixed order and size, the extremal graphs with the maximum Aα-index are the double nested graphs, i.e., bipartite chain graphs. On the other hand, we establish a series of inequalities respecting the entries of the Perron vector of Aα(G) of double nested graphs. Then we obtain some lower and upper bounds on the Aα-index of double nested graphs using the eigenvector and matrix techniques. Finally, we provide some computational results to compare these bounds.