graph theory based methods are powerful means for representing structural systems so that their geometry and topology can be understood clearly. the combination of graph theory based methods and some metaheuristics can offer effective solutions for complex engineering optimization problems. this paper presents a charged system search (css) algorithm for the free shape optimizations of thin-wall...

Suppose $X$ is a simple graph. The $X-$join $Gamma$ of a set ofcomplete or empty graphs ${X_x }_{x in V(X)}$ is a simple graph with the following vertex and edge sets:begin{eqnarray*}V(Gamma) &=& {(x,y) | x in V(X) & y inV(X_x) },\ E(Gamma) &=& {(x,y)(x^prime,y^prime) | xx^prime in E(X) or else x = x^prime & yy^prime in E(X_x)}.end{eqnarray*}The $X-$join graph $Gamma$ is said to be re...

We consider the problem of traveling the contour of the set of all points that are within distance 1 of a connected planar curve arrangement P, forming an embedding of the graph G. We show that if the overall length of P is L, there is a closed roundtrip that visits all points of the contour and has length no longer than 2L + 2n. This result carries over in a more general setting: if R is a com...

A watchman’s walk in a graph G = (V,E) is a minimum closed dominating walk. In this paper, it is shown that the number of vertices in a watchman’s walk on the block intersection graph of a Steiner triple system is between v−3 4 and v−7 2 , for admissible v ≥ 15. Included are constructions to build a design that achieves the minimum bound for any admissible v.

A global offensive alliance in a graph G = (V,E) is a subset S of V such that for every vertex v not in S at least half of the vertices in the closed neighborhood of v are in S. We give an upper bound on the global offensive alliance number of a graph in terms of its degree sequence. We also study global offensive alliances of random graphs.

W e consider random graphs with edge probability Pn-Q, where n is the number of vertices of the graph, /3 > 0 is fixed, and a = 1 or a = (I + 1)/1 for some fixed positive integer I . We prove that for every firstorder sentence, the probability that the sentence is true for the random graph has an asymptotic limit. Also, there is an eflective procedure for generating the value of the limit in cl...

A global offensive alliance in a graph G = (V,E) is a subset S of V such that for every vertex v not in S at least half of the vertices in the closed neighborhood of v are in S. We give an upper bound on the global offensive alliance number of a graph in terms of its degree sequence. We also study global offensive alliances of random graphs.

Not every graph has an Eulerian tour. But every finite, strongly connected graph has a multi-Eulerian tour, which we define as a closed path that uses each directed edge at least once, and uses edges e and f the same number of times whenever tail(e) = tail(f). This definition leads to a simple generalization of the BEST Theorem. We then show that the minimal length of a multi-Eulerian tour is b...

The purpose of this paper is the study of Cayley graph associated to a semihypergroup(or hypergroup). In this regards first  we associate a Cayley graph to every semihypergroup and then we study theproperties of this graph, such as  Hamiltonian cycles in this  graph.  Also, by some of examples we will illustrate  the properties and behavior of  these Cayley  graphs, in particulars we show that ...