Discrete Mathematics

A point determining graph is defined to be a graph in which distinct non adjacent points have distinct neighborhoods. If in addition any two distinct points have distinct closed neighborhoods, it is called point distinguishing graph. A graph G is said to be one point determining, if for any two distinct vertices v_1 and v_2 ?N(v?_1)and? N(v?_2) have at most one vertex in common. A graph G is said to be two point distinguishing if for any two distinct vertices v_1 and? v?_2, the closed neighborhood ? N[v?_1] and ? N[v?_2] , have at most two vertices in common. Here we focus on some properties of one point determining and two point distinguish- ing graphs.

In this work, a class of new computational methods for solution of variable coefficients partial differential equation was developed at step numbers j = 2; resulting into a Trapezoidal rule spectral based computational scheme as reported in Lambert (1973). The accuracy, consistency, stability and convergence properties of these methods were determined. The methods were implemented on some sampled problems that involve both constant and, variable coefficients parabolic partial differential equations; and evaluated by comparing them with some existing difference methods. The results obtained are found to be more rapidly converging as the step lengths h and k approaches zeros. This work provides better alternative numerical solutions to a class of dynamical problems having time dependent boundary conditions. Higher ordered parabolic partial differential equations with defined theoretical solutions to given boundary conditions can be solved directly using this method. 

Every fuzzy context free language L(G) can be generated by N-triangular form of fuzzy context free grammar is Proved in this paper with illustrated examples.

Min-li Yu [1] gives the necessary and sufficient conditions on path factorization of complete multipartite graphs and earlier for path factorization of complete bipartite graph Kazuhiko Ushio [2] gave the necessary and sufficient conditions for the existence of the P_k-design when k is odd. However, for the odd value of k the path factorization problem of complete bipartite graphs i.e. P_k-factorization of complete bipartite graphs, have been studied by many number researchers[3,4,5,6,7]. For any positive integer p, the necessary and sufficient conditions for the existence of the P_2p-factorization of a complete bipartite graph were studied by Hong Wang[8]. Beiliang Du [9] extended the work of Hong Wang [8] and gave the necessary and sufficient conditions for the existence of P_2k-factorization of the complete bipartite multigraphs. In path factorization of complete bipartite symmetric digraphs B. Du [10] already discussed the necessary and sufficient conditions for the existence of P ?_3-factorization of complete bipartite symmetric digraphs. Here in this paper, we will discuss necessary and sufficient conditions for the existence of P ?_2k- factorization of complete bipartite symmetric digraphs, and also in this paper, we construct the P ?_2k- factorization of complete bipartite symmetric digraphs K_(m,n)^*.

In this paper, we introduce the concept of clique covering of a graph. Every vertex covering set, which contains all isolated vertices, is a clique covering set in a graph. We consider the effect of removing a vertex from the graphs on the clique covering number of the graph. We also define well clique covered graphs and prove some related result.

For any graph G, let V(G) and E(G) denote the vertex set and edge set of G respectively. The Boolean function graph B(Kp, INC,?Kq) of G is a graph with vertex set V(G)?E(G) and two vertices in B(Kp, INC,?Kq) are adjacent if and only if they correspond to two adjacent vertices of G, two nonadjacent vertices of G or to a vertex and an edge incident to it in G, For brevity, this graph is denoted by?B4(G). In this paper, structural properties of the complement?B4(G) of B4(G) including eccentricity properties are studied. Also, domination number and neighborhood number are found.

Today software quality is an consequential parameter and to produce high quality reliable software and to keep a specific time schedule is a big challenge. There are many software reliability models that are based on the times of occurrences of errors in the debugging of software. To overcome the challenge many methodology and practices of software engineering have been evolved for developing reliable software. Improving algorithms of directing the manner of software development are underway. Order statistics are used in a broad category of exercised assignments their application in improvisation problems, determination of outliers, linear estimation. In this paper we proposed the detecting the exponential reliable software using SPRT on time domain data based on Ordered Statistics approach. The unknown parameters can be estimated using the Maximum Likelihood Estimation (MLE).

In this paper we discuss the concept of total edge domination in graphs. We prove that for any connected (p,q) – graph G with ? < q - 1, t q - ? where ? denotes the maximum degree of an edge in G and characterize trees and unicyclic graphs which attain this bound. We also prove that t (S(G)) for any connected graph G. We also determine the value of total edge domatic number for some families of graphs.

In this paper we produce a hypergraph using the Peterson hypergraph. We prove that two vertices in this hypergraph are adjacent if and only if they are not adjacent in the Peterson graph. We find maximum independent sets, maximum stable sets, domination number and H-domination number of this hypergraph.

A subset D of the vertex set V(G) of a graph G is said to be a dominating set if every vertex not in D is adjacent to atleast one vertex in D. A dominating set D is said to be an eccentric dominating set if for every v ? V?D, there exists atleast one eccentric point of v in D. A subset D is a distance closed set of G if for each vertex u ? D and for each w ? V?D, there exists atleast one vertex v ? D such that d(u, v) = dG(u, w). An eccentric dominating set D of G is a distance closed eccentric dominating set if the induced subgraph is distance closed. The minimum of the cardinalities of the distance closed eccentric dominating set is called as the distance closed eccentric domination number ?dced(G) of G. In this paper, bounds for ?dced(G), its exact value for some particular classes of graphs and some results on distance closed eccentric domination number are obtained.