TOUGHNESS, FRACTIONAL EXTENDABILITY AND DISTANCE SPECTRAL RADIUS IN GRAPHS

A graph G is said to be t-tough if S >= t <middle dot> c(G-S) for every subset S subset of V (G) with c(G-S) >= 2, where c(G-S) denotes the number of connected components in G-S. A graph G is fractional k-extendable if every k-matching in G can be extended to a fractional perfect matching of G. In this paper, we first establish an upper bound on the distance spectral radius of G to ensure that G is a t1-tough graph. Then we give an upper bound on the distance spectral radius of G to guarantee that G is a t-tough graph. Finally, we show an upper bound on the distance spectral radius of G to guarantee that G is a fractional k-extendable graph.