We generalize to n steps the notion of exact 2-step domination introduced by Chartrand, et al in [2] and suggest a related minimization problem for which we nd a lower bound. A graph G is an exact n-step domination graph if there is some set of vertices in G such that each vertex in the graph is distance n from exactly one vertex in the set. We prove that such subsets have order at least blog2 ...

A group G of permutations of a set is primitive if it acts transitively on , and the only G-invariant equivalence relations on are the trivial and universal relations. A digraph is primitive if its automorphism group acts primitively on its vertex set, and is infinite if its vertex set is infinite. It has connectivity one if it is connected and there exists a vertex α of , such that the induced...

A planar 3-connected graph G is called essentially 4-connected if, for every 3-separator S, at least one of the two components of G−S is an isolated vertex. Jackson and Wormald proved that the length circ(G) of a longest cycle of any essentially 4-connected planar graph G on n vertices is at least 2n+4 5 and Fabrici, Harant and Jendrol’ improved this result to circ(G) ≥ 1 2 (n+ 4). In the prese...

An edge e of a k-connected graph G is said to be a removable edge if G a e is still k-connected, where Ga e denotes the graph obtained from G by deleting e to get G− e and, for any end vertex of e with degree k− 1 in G− e, say x, deleting x and then adding edges between any pair of non-adjacent vertices in NG−e(x). Xu and Guo [28] proved that a 5-connected graph G has no removable edge if and o...

ÐGiven a graph G ˆ …V ;E† with n vertices and m edges, the k-connectivity of G denotes either the k-edge connectivity or the k-vertex connectivity of G. In this paper, we deal with the fully dynamic maintenance of k-connectivity of G in the parallel setting for k ˆ 2; 3. We study the problem of maintaining k-edge/vertex connected components of a graph undergoing repeatedly dynamic updates, such...

let $g$ be a non-abelian group. the non-commuting graph $gamma_g$ of $g$ is defined as the graph whose vertex set is the non-central elements of $g$ and two vertices are joined if and only if they do not commute.in this paper we study some properties of $gamma_g$ and introduce $n$-regular $ac$-groups. also we then obtain a formula for szeged index of $gamma_g$ in terms of $n$, $|z(g)|$ and $|g|...

BACKGROUND A common control condition for transcranial magnetic stimulation (TMS) studies is to apply stimulation at the vertex. An assumption of vertex stimulation is that it has relatively little influence over on-going brain processes involved in most experimental tasks, however there has been little attempt to measure neural changes linked to vertex TMS. Here we directly test this assumptio...

We study M-alternating Hamilton paths, and M-alternating Hamilton cycles in a simple connected graph G on ν vertices with a perfect matchingM . Let G be a bipartite graph, we prove that if for any two vertices x and y in different parts of G, d(x)+d(y) ≥ ν/2+2, then G has an M-alternating Hamilton cycle. For general graphs, a condition for the existence of an M-alternating Hamilton path startin...

The distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r ≥ 2, there exists a connected graph H such that G is the antimedian and J the median subgraphs of ...

Given a simple connected graph G, let K(n) [2(n)] be the minimum cardinality of a set of vertices [edges], if any, whose deletion disconnects G and every remaining component has more than n vertices. For instance, the usual connectivity and the superconnectivity of G correspond to x(0) and ~c(1 ), respectively. This paper gives sufficient conditions, relating the diameter of G with its girth, t...