The last theorem is the highlight of the paper. In the graph theory, two graphs are said to be isomorphic if there is a one-one, onto mapping defined between their set of vertices so as to preserving the adjacency between vertices. An isomorphism defined on a vertex set of a graph to itself is called automorphism of the given graph. Two vertices in a graph are said to be similar if there is an automorphism defined on its vertex set mapping one vertex to the other. In this paper, it has been discussed that every such automorphism defines an equivalence relation on the set of vertices and the number of equivalence classes is same as the number of rotations that the automorphism makes on the vertex set.

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The set of all automorphisms of a graph is a permutation group under the composition of permutations. This group is called automorphism group of the graph. A graph is said to be vertex transitive if its automorphism group acts transitively on its vertex set. The path degree sequence of a vertex in a graph is the list of lengths of paths having this particular vertex as initial vertex. The ordered set of all such sequences is called path degree sequence of the graph.

## Read e-book Mathematical Combinatorics (International Book Series), Vol. 4, 2009

It is conjectured that two graphs are isomorphic iff they have same path degree sequence. In this paper, it has been discussed that this conjecture holds good when both the graphs are vertex transitive. The notion of functional graph has been introduced in this paper. The functional graph of any two isomorphic graphs is a graph in which the vertex set is the union of vertex sets of isomorphic graphs and two vertices are connected by an edge iff they are connected in any one of the graph when they belong to the same graph or one vertex is the image of the other under the given isomorphism when they are in different set of vertices.

It has been proved that the functional graphs obtained from two isomorphic complete bipartite graphs are vertex transitive. Keywords : graph automorphism ; functional graph ; vertex transitive graph ; path degree sequence. Note on Combinatorics. Authors: Choe Ryujin Comments: 6 Pages. Simple solution of traveling salesman problem Category: Combinatorics and Graph Theory. In this paper, motivated by the notion of generalized single valued neutrosophic graphs of first type, we defined a new neutrosophic graphs named generalized interval valued neutrosophic graphs of first type GIVNG1 and presented a matrix representation for it and studied few properties of this new concept.

In this article, we present an algorithm method for finding the shortest path length between a paired nodes on a network where the edge weights are characterized by single valued triangular neutrosophic numbers. The proposed algorithm gives the shortest path length from source node to destination node based on a ranking method. Finally a numerical example is presented to illustrate the efficiency of the proposed approach Category: Combinatorics and Graph Theory.

In this paper, we introduced a new neutrosophic graphs called complex neutrosophic graphs of type1 CNG1 and presented a matrix representation for it and studied some properties of this new concept. Mathar Comments: Pages are adjacency matrices for connected simple graphs up to 7 nodes.

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We create the unlabeled or vertex-labeled graphs with up to 10 edges and up to 10 vertices and classify them by a set of standard properties: directed or not, vertex-labeled or not, connectivity, presence of isolated vertices, presence of multiedges and presence of loops. We present tables of how many graphs exist in these categories.

Frempong Comments: 12 Pages. The traveling salesman can determine by hand, with zero or negligible error, the shortest route from home base city to visit once, each of three cities, 10 cities, 20 cities, cities, or cities, and return to the home base city. The general approach to solving the different types of NP problems is the same, except that sometimes, specific techniques may differ from each other according to the process involved in the problem.

In the salesman problem, the first step is to arrange the data in the problem in increasing order, since one's interest is in the shortest distances. The main principle here is that the shortest route is the sum of the shortest distances such that the salesman visits each city once and returns to the starting city. The approach in this paper is different from the author's previous approach viXra In this paper, one starts with only the least ten distances and only if a needed distance is not among the set of the least ten distances, would one consider distances greater than those in the set of the ten least distances.

The shortest route to visit nine cities and return to the starting city was found in this paper. It was also found out that even though the length of the shortest route is unique, the sequence of the cities involved is not unique. The approach used in this paper can be applied in workforce project management and hiring, as well as in a country's workforce needs and immigration quota determination.

Since an approach that solves one of these problems can also solve other NP problems, and the traveling salesman problem has been solved, all NP problems can be solved, provided that one has an open mind and continues to think. Authors: Alexey V. Komkov Comments: 2 Pages. These certificates establish the best currently known lower bounds of the numbers Van der Waerden W 7, 3 , W 8, 3 , W 9, 3 , W 10, 3 , W 11, 3. In this paper, we first define the concept of neutrosophic soft expert graph. We have established a link between graphs and neutrosophic soft expert sets.

Basic operations of neutrosophic soft expert graphs such as union, intersection and complement are defined here. The concept of neutrosophic soft expert soft graph is also discussed in this paper. Finally, an illustrative example is given and a comparison analysis is conducted between the proposed approach and other existing methods, to verify the feasibility and effectiveness of the developed approach.

Komkov Comments: Pages. Genetic algorithm is a good tool for finding the global minimum in many discrete problems. In particular, it has proven itself in problems where there is no any apriori information about the possibilities of narrowing the search, or the specifics of the problem do not involve such. This work describes the procedure of using a genetic algorithm as applied to the search of van der Waerden numbers. Some new lower bounds of van der Waerden numbers were found using this procedure.

Salama Comments: 8 Pages.

## Franco V Saliola -- Publications

This paper is devoted for presenting new neutrosophic similarity measures between neutrosophic graphs. We proposetwo ways to determine the neutrosophic distance between neutrosophic vertex graphs. The two neutrosophic distances are based on the Haussdorff distance,and a robust modified variant of the Haussdorff distance, moreover we show that they both satisfy the metric distance measure axioms. Furthermore, a similarity measure between neutrosophic edge graphs that is based on a probabilistic variant of Haussdorff distance is introduced. The aim is to use those measures for the purpose of matching neutrosophic graphs whose structure can be described in the neutrosophic domain.

Authors: Eman. Elghawalby , A. The aim of this paper is to introduce a new approach to Mathematical Morphology based on neutrosophic set theory. Basic definitions for neutrosophic morphological operations are extracted and a study of its algebraic properties is presented.

In our work we demonstrate that neutrosophic morphological operations inherit properties and restrictions of Fuzzy Mathematical Morphology Category: Combinatorics and Graph Theory. The subdivision graph S G of a graph G is the graph obtained by inserting a new vertex into every edges of G.

Authors: K. Reji Kumar , Renny P. Varghese Comments: 10 Pages. The Duplication graph DG of a graph G, is obtained by inserting new vertices corresponding to each vertex of G and making the vertex adja- cent to the neighbourhood of the corresponding vertex of G and deleting the edges of G. In this paper we determine the A - spectra and L - spectra of the two new joins of graphs for a regular graph G1 and an arbitrary graph G2.

As an application we give the number of spanning tree, the Kirchhoff index and Laplace energy like invariant of the new join. Also we obtain some infinite family of new class of integral graphs Category: Combinatorics and Graph Theory. We present a spectral theory of uniform, regular and linear hyper- graph. The main result are the nature of the eigen values of k; r - regular linear hypergraph and the relation between its dual and line graph.

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We also discuss some properties of Laplacian spectrum of a k; r - regular hypergraphs. Authors: Damodar Rajbhandari Comments: 9 Pages. This paper will help you to understand, "How Euler solved this problem?

And, It will give some intuition of "Topology" and "Graph Theory". This is a numerical study of the combinatorial problem of packing hexagons of some equal size into a larger hexagon. The problem is well defined if all hexagon edges have integer length and if their centers and vertices share the common lattice points of a triangular grid with unit distances. Authors: Lucas Allen Comments: English, 9 pages, matrix equations and examples.

This article presents an algorithm for solving the graph isomorphism problem. Under certain circumstances the algorithm is definitely polynomial time, and it could possibly always be polynomial time, but that hasn't been verified. The algorithm also hasn't been tested on graphs with more than three nodes, nor has it been reviewed by anyone so far. Authors: Michail Zak Comments: 15 Pages. The challenge of this paper is to relate artificial intuition-based intelligence, represented by self-supervised systems, to solutions of NP-complete problems.

By self-supervised systems we understand systems that are capable to move from disorder to order without external effort, i. It has been demonstrated, [1], that such systems exist in the mathematical world: they are presented by ODE coupled with their Liouville equation, but they belong neither to Newtonian nor to quantum physics since they are capable to violate the second law of thermodynamics. That suggests that machines could not simulate intuition-based intelligence if they are composed only of physical parts, but without digital components.

Nevertheless it was found such quantum-classical hybrids, [1], that simulates some of self-supervised systems. The main achievement of this work is a demonstration that self-supervised systems can solve NP-complete problems in polynomial time by replacing an enumeration of exponentially large number of possible choices with a short cut provided by a non-Newtonian and non-quantum nature of self-supervised systems. Authors: Marius Coman Comments: Pages. The purpose of this book is to show that the method of concatenation can be a powerful tool in number theory and, in particular, in obtaining possible infinite sequences of primes.

Authors: Linfan Mao Comments: Pages. The International J. Authors: B. Chaluvaraju , H. Boregowda , S. Kumbinarsaiah Comments: 6 Pages.