On the Da-spectral radius of cacti and bicyclic graphs

The D alpha-matrix of a connected graph G is defined as D alpha (G) = alpha Tr(G) + (1 - alpha)D(G), where 0 < alpha < 1, Tr(G) is the diagonal matrix of the vertex transmissions of G and D(G) is the distance matrix of G. The largest eigenvalue of D alpha(G) is called the D alpha-spectral radius of G. In this paper, we determine the unique graph with minimum D alpha-spectral radius among the cacti of order n with k (k & GE; 1) cycles and at least one pendent vertex. We also show that the D alpha-spectral radius of Bn* (the graph obtained from a star of order n by adding two nonadjacent edges) is less than the D alpha-spectral radius of the most of bicyclic graphs of order n.(c) 2022 Elsevier B.V. All rights reserved.