The crossing number $\mathrm{cr}(G)$ of a graph $G$ is the minimum edge crossings over all drawings in plane. main aim this paper to give numbers join products eight graphs on six vertices with paths and cycles $n$ vertices. proofs are done help several well-known auxiliary statements, idea which extended by suitable classification subgraphs that do not cross edges examined graphs.

Shepherd (1995) proved that the stable set polytopes of near-bipartite graphs are given by constraints associated with the complete join of antiwebs only. For antiwebs, the facet set reduces to rank constraints associated with single antiwebs by Wagler (2004b). We extend this result to a larger graph class, the complements of fuzzy circular interval graphs, recently introduced in Chudnovsky and...

We prove that the ribbon graph polynomial of a embedded in an orientable surface is irreducible if and only neither disjoint union nor join graphs. This result analogous to fact Tutte connected non-separable.

For fine grain task graphs, duplication-based scheduling algorithms are generally more efficient than list and cluster-based algorithms. However, most duplication-based heuristics try to duplicate all possible ancestor nodes of a given join node, in order to reduce the earliest start time (EST) of the join node, even though these ancestor nodes have already been allocated in previous steps. Thu...

The distance between two vertices of a simple connected graph G, denoted as d(u,v), is the length shortest path from u to v and always symmetrical. An ordered subset W=w1,w2,w3,⋯,wk V(G) resolving set for if ∀u,v∈V(G), there exists wi∈W ∋ d(u,wi)≠d(v,wi). A with minimal cardinality called metric basis. dimension G basis dim(G). For G1=(V1,E1,) G2=(V2,E2), their join by G1+G2. vertex G1+G2 V1∪V2...

Subgraph enumeration aims to find all the subgraphs of a large data graph that are isomorphic to a given pattern graph. As the subgraph isomorphism operation is computationally intensive, researchers have recently focused on solving this problem in distributed environments, such as MapReduce and Pregel. Among them, the state-of-the-art algorithm, TwinTwigJoin, is proven to be instance optimal b...

We consider the following conjecture: Let G be a k-regular simple graph with an even number n of vertices. If k ≥ n/2 then G is k-edge-colourable. We show that this conjecture is true for graphs that are join of two graphs and we provide a polynomial time algorithm for finding a k-edge-colouring of these graphs. c © 2007 Elsevier B.V. All rights reserved.

We call a graph G O-graph if there is an optimal coloring of the set of vertices and an optimal (disjoint) covering with cliques such that any class of colors intersects any clique. In this paper, it has been established the relation to [p, q, r]-partite graphs and the fact that the O-graphs admit a k-clique-join.

Graphs are the central objects to study during our course. The concept of graphs is simple but it has found very wide application. It can be used to model many real-world networks, such as the internet. This is a “pure mathematics” module, and hence we will focus on the mathematical theory of graphs, rather than on the applications. Informally, a graph is a collection of points (called “vertice...