International Journal of Innovative Research and Development

Abstract

In mathematics and computer science, graph theory is the study of graphs. Mathematical structures are used to model pairwise relations between objects from a certain collection. A graph in this context refers to a collection of vertices or nodes and a collection of edges that connect pairs of vertices. A graph may be undirected, which means that there is no distinction between the two vertices associated with each edge or its edges may be directed from one vertex to another vertex. The development of graphs therefore of major interest in computer science. In this paper , for mathematical computing, the two search methods namely BFS, DFS are thoroughly analyzed by means of  definitions, analysis of the algorithms followed by findings and observations.

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Let k  be an  infinite field,  I an infinite set,  V be  a  k_-Vector-space, and  g :k^I  → V  a    k-linear map.  It is shown that if  dim_k(V) is not too  large (under various hypotheses on  card(k)and  card(I),  if it is finite, respectively less than  card(k),  respectively less than the continuum), then ker(g)must contain elements  (u_i)_i∈I   with all but finitely many components  u_i  nonzero. These results are used to prove that every homomorphism from a direct product    Π _IA_iof not-necessarily associative  algebras  A_i  onto an  algebra  B,  where  dim_k(B) is not too  large (in the same senses)  is the sum of a map factoring through the projection Π _IA_ionto the product of finitely many of the  A_i,  and a map into the ideal  {b∈ B |bB= Bb₌{0}} ⊆ B.Detailed consequences are noted in the case where the  A_iare Lie algebras. A version of the above result is also obtained with the field  k  replaced by a commutative valuation ring. This note resembles in that the two papers obtain similar results on homomorphisms on infinite product algebras; but the methods are different, and the hypotheses under which the methods of one note work are in some ways stronger, in others weaker, than those of the other.  Also, in  we obtain many consequences from our results, while here we aim for brevity, and after one main result about general algebras, restrict ourselves to a couple of quick consequences for Lie algebras. .

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This paper presents notions of 1- and 2-connectivity. It starts with 1-connectivity of directed graphs, and it then examines 2-connectivity of undirected graphs. Depth-first search is the method of choice to calculate low order connectivity information.  The algorithms which are designed for connectivity properties are originally due to Tarjan [1]. This paper follows the path-based development of [2], which simplifies the algorithms to eliminate the depth-first spanning tree.

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