Discrete Mathematics

A matroid is a set with an independent structure defined on it. A matroid abstracts and generalizes the notion of linear independence in vector spaces and independence in graphs. Matroids unite the concepts of graph theory, linear algebra, projective geometry, transversal theory, and combinatorial optimization. Applications of matroids involve different areas such as combinatorial optimization, network theory, coding theory and many other areas. Matroids can be found in projective geometry; the fano plane of order 2 gives rise to a matroid. An important application of matroids in optimization involves the greedy algorithm. Kruskal’s algorithm for ?nding a minimal spanning tree which is an example of the greedy algorithm can be used to understand how matroids can be involved in the greedy algorithm. Consider a network of vertices with weighted links between the vertices. Our goal is to ?nd a collection of links that connect all vertices using the smallest weight. That is a spanning tree with minimal weights. Kruskal’s algorithm can be generalized to a matroid by taking a matroid M and a function w:M?R which assigns weights to each element. The goal is to ?nd the basis B of M such that ?w(x) where x?B is minimized. The greedy algorithm is a characterization of the matroid. Matroids are the structures in which the greedy algorithm works successfully

Number is the language of science and the best way to express it is symbols, and numerals are forms(symbols) written by the numbers codes. Alsumariun expressed Semites for numbers letters alphabet, and the Babylonians expressed them in cuneiform, and Egyptians wrote numbers in the form of horizontal and vertical lines and fees hieroglyphics and the Greeks used first letters of words to denote numbers, as did the Phoenicians and Hebrews, while Romans used vertical lines and then evolved to take the form of letters of the alphabet. Arabs before Islam expressed numbers in terms of Alphabet and after the advent of Islam and the descent of the Holy Quran and the receipt of a lot of numbers in it written in Arabic language Muslims expressed numbers writing method instead of symbols.

The independent dominating set problem is a classical optimization problem and has been shown to be NP-Complete. This study finds a molecular computing model to solve the independent dominating set problem, based on Adleman-Lipton model. It proves how to apply stickers in the sticker based model to construct the DNA solution space of the independent dominating set problem and how to apply DNA operations in the Adleman-Lipton model to solve that problem from the solution space of stickers. The time complexity of the proposed computational model is O(n + 2m) and to verify this model, a small independent dominating set problem was solved. This proves the capacity of molecular computing for solving the complex independent dominating set problem.

A Total Dominating Color Transversal Set of a Graph G is a Total Dominating Set which is also Transversal of Some ? - Partition of vertices of G. Here ? is the Chromatic number of the graph G. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. In this paper we find this number for Generalized Wheel Graph, Petersen Graph, Herschel Graph, Grotszch graph and Helm Graph.

A separation cordial labeling of graph is a bijection f from to such that each edge uv is assigned the label 1 if ) is an odd number and label 0 if is an even number. Then the number of edges labeled 0 and the number of edges labeled 1 differ by at most 1. A graph has a separation cordial labeling, then it is called separation cordial graph. Here, the bistar the splitting graphs of and , the shadow graph of and square graph of are discussed and found to be separation cordial.

A tree T is called starlike [2], if it contains a vertex v for which deg (v) ? 3 and all other vertices of T have degree 1 or 2. If deg (v) = k, the starlike tree T is k – branched and T – v has k components, each of which are trees. In this paper, we characterize the 3 – branched starlike spanning trees of a given two dimensional mesh M(m, n) m, n ? 3 and then find its number with junction vertex [2] of degree 3.

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)^*.