Abstract
The Dα-matrix of a connected graph G is defined asDα(G)=αTr(G)+(1−α)D(G), where 0≤α<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 Dα(G) is called the Dα-spectral radius of G. In this paper, we determine the unique graph with minimum Dα-spectral radius among the cacti of order n with k (k≥1) cycles and at least one pendent vertex. We also show that the Dα-spectral radius of Bn⁎ (the graph obtained from a star of order n by adding two nonadjacent edges) is less than the Dα-spectral radius of the most of bicyclic graphs of order n.
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