A neighborhood total dominating set in a graph G is a dominating set S of G with the property that the subgraph induced by N(S), the open neighborhood of the set S, has no isolated vertex. The neighborhood total domination number γnt(G) is the minimum cardinality of a neighborhood total dominating set of G. Arumugam and Sivagnanam introduced and studied the concept of neighborhood total dominat...

The notion of a p-competition graph and the p-competition number of a graph were introduced by S. -R. Kim, T. A. McKee, F. R. McMorris, and F. S. Roberts as a generalization of a competition graph and the competition number of a graph, respectively. Let p be a positive integer. The p-competition graph Cp(D) of a digraph D = (V, A) is a (simple undirected) graph which has the same vertex set V a...

We take G to be an undirected graph without loops or multiple edges, with vertex set V (G) = f1; : : : ; ng, and with (0; 1)-adjacency matrix A. Let P denote the orthogonal projection of IR onto the eigenspace E( ) of A, and let fe1; : : : ; eng be the standard orthonormal basis of IR. Since E( ) is spanned by the vectors Pej (j = 1; : : : ; n) there exists X V (G) such that the vectors Pej (j ...

For any graph G, let V (G) and E(G) denote the vertex set and the edge set of G respectively. The Boolean function graph B(G, L(G),NINC) of G is a graph with vertex set V (G) ∪ E(G) and two vertices in B(G, L(G),NINC) are adjacent if and only if they correspond to two adjacent vertices of G, two adjacent edges of G or to a vertex and an edge not incident to it in G. For brevity, this graph is d...

Let G be a graph of order n and circumference c(G). Let G be the complement of G. We prove that max{c(G), c(G)} ≥ d 2n 3 e and show sharpness of this bound. © 2008 Elsevier B.V. All rights reserved.

Motivated by some computations of Feynman integrals and certain conjectures on mixed Tate motives, Bejleri Marcolli posed questions about the $\mathbb{F}_1$-structure (in sense torification) complement a hyperplane arrangement, especially for an arrangement defined in space cycles graph. In this paper, we prove that has if only it is Boolean. We also cycle graph Boolean basis consisting such an...

Given a simple graph Γ, we describe a “lifting” to a 3-uniform hypergraph φ(Γ) that sends the complement of Γ to the complement of φ(Γ). We consider the effects of this lifting on cycles, complete subhypergraphs, and complete subhypergraphs missing a single hyperedge. Our results lead to natural lower bounds for some hypergraph Ramsey numbers.

We show that given a trivalent graph in S, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful planar surface. Thin position for a knot is a powerful tool developed by Gabai [1]...

A graph G is self complementary if it is isomorphic to its complement G. In this paper we define bipartite self-complementary graphs, and show how they can be used to understand the structure of self-complementary graphs. For G a selfcomplementary graph of odd order, we describe a decomposition of G into edge disjoint subgraphs, one of which is a bipartite self-complementary graph of order |G|−...

For a finite group G, the proper power graph P∗(G) of G is the graph whose vertices are non-trivial elements of G and two vertices u and v are adjacent if and only if u 6= v and um = v or vm = u for some positive integer m. In this paper, we consider the complement of P∗(G), denoted by P(G). We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a ...