Abstract

For a connected graph G and alpha is an element of[0,1], let D-alpha(G) be the matrix D-alpha(G) = alpha Tr(G) + (1 - alpha)D(G), where D(G) is the distance matrix of G and Tr(G) is the diagonal matrix of its vertex transmissions. Let K-m be a complete graph of order m. For n, s fixed, n > s, let G(p) = K-s boolean OR (K-p boolean OR Kn-s-p) be the graph obtained from K-s and K-p boolean OR Kn-s-p and the edges connecting each vertex of K-s with every vertex of K-p boolean OR Kn-s-p. This paper presents some extremal results on the spectral radius of D-alpha(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 (alpha) under bar is an element of(3/4, 3n-s/4n-s) such that if alpha is an element of[0, (alpha) under bar) then the minimal spectral radius of D-alpha(G) is uniquely attained by G = G(1) 2. there exists a unique (alpha) over bar is an element of(3/4, 3n-s/4n-s), (alpha) over bar >= (alpha) under bar, such that if alpha is an element of((alpha) over bar, 1) then the minimal spectral radius of D-alpha(G) is uniquely attained by G = G left perpendicular n-s/2 right perpendicular, and 3. if alpha = 1 then the minimal spectral radius of Tr(G) is n - 1 + inverted right perpendicular n-s/2 inverted left perpendicular and it is uniquely attained by G = G left perpendicular n-s/2 right perpendicular. Furthermore, in terms of n and s, a tight lower bound l(n, s) of (alpha) under bar and a tight upper bound u(n, s) of (alpha) over bar are obtained. Finally, for s fixed, it is observed that lim n ->infinity l(n, s) = lim n ->infinity (alpha) under bar = lim n ->infinity u(n, s) = lim n ->infinity (alpha) over bar = 3/4. (C) 2019 Elsevier Inc. All rights reserved.