Let G= (V, E) be a connected simple graph. A labeling f: V®Z2 induces an edge labeling f*: E®Z2 defined by f* (xy) =f (x)+f (y) for each xy Î E. For i Î Z2, let vf (i) =|f-1 (i)| and ef (i) =|f*-1 (i)|. A labeling f is called friendly if |vf (1) - vf (0)|£1. The full friendly index set of G consists all possible dierences between the number of edges labeled by 1 and the number of edges labeled by 0. In recent years, full friendly index sets for certain GRAPHS were studied, such as tori, grids P2´Pn, and CYLINDERs Cm´Pn for some n and m. In this paper we study the full friendly index sets of CYLINDER GRAPHS Cm´P2 for m ³ 3, Cm´P3 for m ³4 and C3´Pn for n ³ 4. The results in this paper complement the existing results in literature, so the full friendly index set of CYLINDER GRAPHS are completely determined.

We determine the forbidden induced subGRAPHS for the intersection of the classes of chordal bipartite GRAPHS and line GRAPHS of acyclic directed GRAPHS. This is a first step towards finding the forbidden induced subGRAPHS for the class of line GRAPHS of directed GRAPHS.

Intersecting cylindrical shells are frequently encountered in various types of engineering structures. Despite their common occurrence, reliable and accurate analytical methods have not been generally available, and consequently accurate design information for such configurations has also been generally lacking. To contribute to the understanding of the problem, a finite element analysis is carried out for two normally intersecting cylindrical shells. Two types of loading have been applied to the structure. Results of stresses are provided, of both the outside and the inside, for the two shells. The overall agreement between the finite element predictions and experimental results is a very good one. It has been shown that by using the finite element method together with a "mesh generation" technique, both computational efficiency and minimum user time can be retained.      

A set W ,V (G) is called a resolving set, if for every two distinct vertices u,v 2 V (G) there exists w 2 W such that d(u, w) 6= d(v, w), where d(x,y) is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we establish a family of graph called Solis graph, and we prove that if G is a minimal edge unique base graph with the base of size two, then G belongs to the Solis GRAPHS family. Finally, an algorithm is given for , nding the metric dimension of a Solis graph.

The Hosoya index $Z(G)$ of a graph $G$ is the total number of matchings in it. In this paper, the recursive formulas of the Hosoya index of semitotal graph $Q(G)$ and total graph $T(G)$ for certain GRAPHS $G$ are obtained. Moreover, we obtain the bounds of the Hosoya index of semitotal and total GRAPHS of a connected graph $G$.

Let $t_p(G)$ denote the number of paths in a graph $G$ and let $f:E\rightarrow \mathbb{Z}^+$ be an edge labeling of $G$. The weight of a path $P$ is the sum of the labels assigned to the edges of $P$. If the set of weights of the paths in $G$ is $\{1,2,3,\dots,t_p(G)\}$, then $f$ is called a Leech labeling of $G$ and a graph which admits a Leech labeling is called a Leech graph. In this paper, we prove that the complete bipartite GRAPHS $K_{2,n}$ and $K_{3,n}$ are not Leech GRAPHS and determine the maximum possible value that can be given to an edge in the Leech labeling of a cycle.

A coalition in a graph G = (V, E) consists of two disjoint sets V1 and V2 of vertices, such that neither V1 nor V2 is a dominating set, but the union V1 , V2 is a dominating set of G. A coalition partition in a graph G of order n = |V| is a vertex partition π,= {V1, V2, …, , Vk} such that every set Vi either is a dominating set consisting of a single vertex of degree n-1, or is not a dominating set but forms a coalition with another set Vj. Associated with every coalition partition π,of a graph G is a graph called the coalition graph of G with respect to π, , denoted CG(G,π, ), the vertices of which correspond one-to-one with the sets V1,V2,…, , Vk of π,and two vertices are adjacent in CG(G,π,) if and only if their corresponding sets in π,form a coalition. In this paper, we initiate the study of coalition GRAPHS and we show that every graph is a coalition graph.

In this paper the p-GRAPHS Matroeids and Hyper GRAPHS and their Matrices are considered. Also it has been shown that a Matroeid and hyper graph are a p-graph and that there is a partial ordered relation between the nodes of the GRAPHS.