On the [formula omitted]-index of minimally 2-connected graphs with given order or size

Abstract

For any real α∈[0,1], Nikiforov defined the Aα-matrix of a graph G as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. The largest eigenvalue of Aα(G) is called the α-index or the Aα-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∈[12,1) among all minimally 2-connected graphs with given order or size, respectively.