Pair Of Distinct Vertices Research Articles

Factorizations of regular graphs

Let G be a k-regular graph of order 2 n such that k≥ n. Hilton ( J. Graph Theory, 9 (1985), 193–196) proved that G contains at least ⌊ k 3 ⌋ edge-disjoint 1-factors. Hilton's theorem is improved in this paper that G contains at least ⌊ k 2 ⌋ edge-disjoint 1-factors. The following result is also proved in this paper: Let G be a 2-connected, k-regular, non-bipartite graph of order at most 3 k − 3 and x, y be a pair of distinct vertices. If Gβ { x, y} is connected, then G contains an ( x, y)-Hamilton path.

An optimal algorithm to find the degrees of connectedness in an undirected edge-weighted graph

Given an undirected graph on n vertices, whose edges are weighted by non-negative real numbers, the degree of connectedness between two distinct vertices is a concept of fundamental importance in the study of undirected fuzzy graphs and symmetrical fuzzy relations. In this paper we obtain an optimal O( n 2) algorithm that finds the degree of connectedness between all pairs of distinct vertices in the graph. It is further shown that the running time of this algorithm is output sensitive.

Isomorphism of graphs which are pairwise k-separable

A polynomial time algorithm for testing isomorphism of graphs which are pairwise k -separable for fixed k is given. The pairwise k -separable graphs are those graphs where each pair of distinct vertices are k -separable. This is a natural generalization of the bounded valence test of Luks. The subgroup of automorphisms of a hypergraph whose restriction to the vertices is in a given Γ k group, for fixed k is constructed in polynomial time.

Connected graphs with a minimal number of spanning trees

The problem studied is the following: Find a simple connected graph G with given numbers of vertices and edges which minimizes the number t μ ( G), the number of spanning trees of the multigraph obtained from G by adding μ parallel edges between every pair of distinct vertices. If G is nearly complete (the number of edges q is ≥( 2 P )− p+2 where p is the number of vertices), then the solution to the minimization problem is unique (up to isomorphism) and the same for all values of μ. The present paper investigates the case where q<( 2 P )− p+2. In this case the solution is not always unique and there does not always exist a common solution for all values of μ. A (small) class of graphs is given such that for any μ there exists a solution to the problem which is contained in this class. For μ = 0 there is only one graph in the class which solves the problem. This graph is described and the minimum value of t 0( G) is found. In order to derive these results a representation theorem is proved for the cofactors of a special class of matrices which contains the tree matrices associated with graphs.

On Spanning and Dominating Circuits in Graphs

AbstractA set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.

On existence and regularity of graphs with certain properties

In the first part we consider such graphs of finite diameter d and finite girth g in which any two vertices of distances are connected by the same number of distinct paths of the length ⩽ d. These graphs are shown to be regulated or many values of d. g and s. In the second part we study the properties of graphs with g⩽2 d+1 in which all pairs of distinct vertices are connected by exactly t=1 distinct paths (for t=1 we get the so-called Moore graphs) We are concerned mainly with the question of existence for various values of d and t.

The square of a block is strongly path connected

A graph is strongly path connected if between each pair of distinct vertices there exist paths of all lengths greater than or equal to the distance between the vertices. It is shown that the squares of both bridgeless connected DT-graphs and blocks are strongly path connected. Thus as a corollary both these classes of graphs are Hamiltonian, Hamiltonian-connected, and vertex and edge pancyclic.

Hypergraphs and abstract polygons

The notion of a graph has recently been generalized to include structures called hypergraphs which have two or more vertices per edge. A hypergraph is called 2-settled if each pair of distinct vertices is contained in at most one edge. A connected 2-settled hypergraph which has at least two edges through each vertex might be called an abstract polygon. Lemma: Every abstract polygon contains a cycle. Shephard and Coxeter have examined certain abstract polygons called regular complex polygons, each of which is denoted by a symbol p {q} r where there are p vertices on each edge and r edges through each vertex. Theorem: The girth of the non-starry regular complex polygon p {q} r is q. Thus, the number q is finally given a simple combinatoric interpretation.

Chromatic Number and Topological Complete Subgraphs

A graph with m(&gt;1) vertices, each pair of distinct vertices connected by an edge, and also a graph obtained from such a graph by the process of subdividing edges through the insertion of new vertices of valency 2, will be denoted by ≪m, o≫. A graph obtained from a graph with m(&gt;2) vertices in which each pair of distinct vertices are connected by an edge, by deleting n (≤ m-1) edges incident with one and the same vertex, and also a graph obtained from such a graph by the process of subdividing edges through the insertion of new vertices of valency 2, will be denoted by ≪m, n≫.