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

For any graph , the lict graph of a graph is the graph whose vertex set is the union of the set of edges and set of cut vertices of in which two vertices are adjacent if and only if corresponding members are adjacent or incident .A set of edges in a graph is called edge dominating set of if every edge in is adjacent to atleast one edge in ,denoted as and is the minimum cardinality of edge dominating set in . In this paper, many bounds on were obtained in terms of vertices , edges and other different parameters of but not in terms of elements of . Further we develop its relation with other different domination parameters.

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.

In this paper we characterized a vertex whose removal increases the perfect domination number of a graph. We also consider the pendent vertices whose removal decreases the perfect domination number.

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 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.

This paper investigates a 2-D discrete time predator - prey interaction model. The equilibrium points are obtained and stability of the equilibrium points is analyzed. The phase portraits are obtained for different sets of parameter values. Also a bifurcation diagram is presented. Numerical simulations are performed and they exhibit rich dynamics of the discrete model.

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.

In this communications, the concept of Pathos Adjacency Line graph PAL(T) of a binary tree T is introduced. We decompose PAL(T) of T in to an edge disjoint complete bipartite subgraphs and then give the reconstruction of T. We also present a characterization of those graphs whose pathos adjacency line graphs are planar, outerplanar, maximal outerplanar, minimally non-outerplanar and non-Eulerian.