Abstract

For a connected graph G and α∈[0,1], let Dα(G) be the matrix Dα(G)=αTr(G)+(1−α)D(G), where D(G) is the distance matrix of G and Tr(G) is the diagonal matrix of its vertex transmissions. Let Km be a complete graph of order m. For n,s fixed, n>s, let Gp=Ks∨(Kp∪Kn−s−p) be the graph obtained from Ks and Kp∪Kn−s−p and the edges connecting each vertex of Ks with every vertex of Kp∪Kn−s−p. This paper presents some extremal results on the spectral radius of Dα(G) that generalize previous results on the spectral radii of the distance matrix and distance signless Laplacian matrix. Among all connected graphs G on n vertices with a vertex/edge connectivity at most s, it is proved that 1. there exists a unique [Formula presented] such that if α∈[0,α_) then the minimal spectral radius of Dα(G) is uniquely attained by G=G1, 2. there exists a unique [Formula presented], α‾≥α_, such that if α∈(α‾,1) then the minimal spectral radius of Dα(G) is uniquely attained by [Formula presented], and 3. if α=1 then the minimal spectral radius of Tr(G) is [Formula presented] and it is uniquely attained by [Formula presented]. Furthermore, in terms of n and s, a tight lower bound l(n,s) of α_ and a tight upper bound u(n,s) of α‾ are obtained. Finally, for s fixed, it is observed that [Formula presented].