On the minimal [formula omitted] spectral radius of graphs subject to fixed connectivity

Abstract

For a connected graph G and α∈[0,1], let Dα(G) be the matrixDα(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 that1.there exists a unique α_∈(34,3n−s4n−2s) 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 α‾∈(34,3n−s4n−2s), α‾≥α_, such that if α∈(α‾,1) then the minimal spectral radius of Dα(G) is uniquely attained by G=G⌊n−s2⌋, and3.if α=1 then the minimal spectral radius of Tr(G) is n−1+⌈n−s2⌉ and it is uniquely attained by G=G⌊n−s2⌋. 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 limn→∞⁡l(n,s)=limn→∞⁡α_=limn→∞⁡u(n,s)=limn→∞⁡α‾=34.