Greedy Forwarding algorithm is a widely-used routing algorithm for wireless networks. However, it can fail if the wireless network topologies contain void scenarios (or simply called voids). Since Yao Graph and Theta Graph are two types of geometric graphs exploited to construct wireless network topologies, this paper studies whether these two types of graphs can contain voids. Specifically, th...

An edge-labeling f of a graph G is an injection from E(G) to the set of integers. The edge-bandwidth of G is B 0 (G) = min f fB 0 (f)g, where B 0 (f) is the maximum diierence between labels of incident edges of G. The m-theta graph (l 1 ; : : : ; l m g is the graph consisting of m pairwise internally disjoint paths with common endpoints and lengths l 1 l m. We determine the edge-bandwidth of al...

We study the relations of knots and links contained in a spatial graph. This is an survey article on the results about knots and links contained in a spatial graph. We do not intend to cover all results in this topic. We only treat some of them here. The set of knots and links contained in a spatial graph is a naive invariant of spatial graph. However it is of course not a complete invariant in...

A generalized-theta-graph is a graph consisting of a pair of end vertices joined by k (k ≥ 3) internally disjoint paths. We denote the family of all the n-vertex generalized-theta-graphs with k paths between end vertices by Θnk . In this paper, we determine the sharp lower bound and the sharp upper bound for the total number of matchings of generalized-theta-graphs in Θnk . In addition, we char...

We generalize the well-known theorem of Corrádi and Hajnal which says that if a given graph G has at least 3k vertices and the minimum degree of G is at least 2k, then G contains k vertex-disjoint cycles. Our main result is the following; for any integer k, there is an absolute constant ck satisfying the following; let G be a graph with at least ck vertices such that the minimum degree of G is ...

‎In this paper‎, ‎first we introduce the notion of $frac{1}{2}$-modular metric spaces and weak $(alpha,Theta)$-$omega$-contractions in this spaces and we establish some results of best proximity points‎. ‎Finally‎, ‎as consequences of these theorems‎, ‎we derive best proximity point theorems in modular metric spaces endowed with a graph and in partially ordered metric spaces‎. ‎We present an ex...

Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimumspanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds—with vanishing error terms—hold for theta gr...

We introduce a generalization of the celebrated Lovász theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial Laplacians, and provides a semidefinite programming upper bound of the independence number of a simplicial complex. We consider properties of the graph theta number such...

A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order n ≥ 9 and size at least b7n−13 2 c contains two disjoint theta graphs. We also show that every 2-edge-connected graph of order n ≥ 6 and size at least 3n− 5 contains two disjoint cycles, such that any specified vertex with degree at least three belongs to...

The generalized theta graph s 1 ;:::;s k consists of a pair of endvertices joined by k internally disjoint paths of lengths s 1 ; : : :; s k 1. We prove that the roots of the chromatic polynomial ((s 1 ;:::;s k ; z) of a k-ary generalized theta graph all lie in the disc jz ? 1j 1 + o(1)] k= logk, uniformly in the path lengths s i. Moreover, we prove that 2;:::;2 ' K 2;k indeed has a chromatic r...