Global Analysis and Discrete Mathematics
Volume:5 Issue: 1, Winter and Spring 2020

The graphs play an important role in our daily life. For example, the urban transport network can be represented by a graph, as the intersections are the vertices and the streets are the edges of the graph. Suppose that some edges of the graph are removed, the question arises, how damaged the graph is. There are some criteria for measuring the vulnerability of graph; the tenacity is the best criteria for measuring it. In this paper, we nd some edge cut sets for organic compounds CnH2n+2 and obtain an upper bound for Te(CnH2n+2) by these edge cut sets.

In this note, we show that cite[Corollary 3.2]{sad} is not always true. In fact, we characterize essential left $phi$-contractibility of the group algebras in terms of compactness of its related locally compact group. Also, we show that for any compact commutative group $G$, $L^{2}(G)$ is always essentially left $phi$-contractible. We discuss the essential left $phi$-contractibility of some Fourier algebras.

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Consider some vertices of a graph G are omitted, there are some criteria for measuring the vulnerability of the graph; Tenacity is one of them. In the definition of tenacity we use vertex cut S and some items, τ (G − S) and ω(G − S), such that τ (G − S) is the number of vertices in the largest component of G − S and ω(G − S) is the number of components of G − S. In this paper we work on tenacity of organic compound CnH2n+2. The graph of this molecule is a tree. We try on tenacity of it by the definition of the tenacity.

In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if f : X × Y × Z −→ W is a bounded tri-linear mapping and h : W −→ S is a bounded linear mapping, then f is regular if and only if hof is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint D^∗∗∗∗ of a tri-derivation D is again tri-derivation.

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Coloring graphs is one of important and frequently used topics in diverse sciences. In the majority of the articles, it is intended to find a proper bound for vertex coloring, edge coloring or total coloring in the graph. Although it is important to find a proper algorithm for graph coloring, it is hard and time-consuming too. In this paper, a new algorithm for vertex coloring, edge coloring and [r, s, t]- coloring is presented. Then, this algorithm is used to solve the applied problems of eight-queens and [r, s, t]- coloring. Here, there are numerical examples to study the efficiency of the method and to compare the results.