On the $α$-index of minimally 2-connected graphs with given order or size

Abstract

For any real α ∈ [0, 1], Nikiforov defined the A α -matrix of a graph G as, where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. The largest eigenvalue of A α (G) is called the α-index or the A α -spectral radius of G. A graph is minimally k-connected if it is k-connected and deleting any arbitrary chosen edge always leaves a graph which is not k-connected. In this paper, we characterize the extremal graphs with the maximum α-index for α ∈ [ 1 2 , 1) among all minimally 2-connected graphs with given order or size, respectively.