The m-order connectivity index (G) m of a graph G is     1 2 1 1 2 1 ... ... 1 ( ) i i im m v v v i i i m d d d  G where 1 2 1 ... i i im d d d  runs over all paths of length m in G and i d denotes the degree of vertex i v . Also,        1 2 1 1 2 1 ... ... 1 ( ) i i im m v v v i i i ms d d d X G is its m-sum connectivity index. A dendrimer is an artificially manufactured or synth...

Let G = (V + s,E) be a graph and let S = (d1, ..., dp) be a set of positive integers with ∑ dj = d(s). An S-detachment splits s into a set of p independent vertices s1, ..., sp with d(sj) = dj , 1 ≤ j ≤ p. Given a requirement function r(u, v) on pairs of vertices of V , an S-detachment is called r-admissible if the detached graph G satisfies λG′(x, y) ≥ r(x, y) for every pair x, y ∈ V . Here λH...

Let d(s) be the smallest number such that every graph of average degree >d(s) contains a subdivision of Ks . So far, the best known asymptotic bounds for d(s) are (1+ o(1))9s2/64 d(s) (1+ o(1))s2/2. As observed by Łuczak, the lower bound is obtained by considering bipartite random graphs. Since with high probability the connectivity of these random graphs is about the same as their average degr...

We show that the directed st-connectivity problem cannot be expressed in symmetric Datalog, a fragment of Datalog introduced in [5]. It was shown there that symmetric Datalog programs can be evaluated in logarithmic space and that this fragment of Datalog captures logspace when augmented with negation, and an auxiliary successor relation S together with two constant symbols for the smallest and...

Gopalan et al. studied in [14] connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on connectivity problems in Schaefer’s framework [26]. A set S of logical relations is Schaefer if all relations in S are either bijunctive, Horn, dual Horn, or affine. They conjectured that the connectivity problem for Schaefer is in P . We disprove their conject...

If G is a graph and n a positive integer, then the generalized Sierpiński graph S G is a fractal-like graph that uses G as a building block. The construction of S G generalizes the classical Sierpiński graphs S n p , where the role of G is played by the complete graph Kp. An explicit formula for the number of connected components in S G is given and it is proved that the (edge-)connectivity of ...