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Smarandache MultiSpace Theory
, 2011
"... Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countr ..."
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Cited by 13 (5 self)
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Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countries appear and each of them has its own system. All of these show that our WORLD is not in homogenous but in multiple. Besides, all things that one can acknowledge is determined by his eyes, or ears, or nose, or tongue, or body or passions, i.e., these six organs, which means the WORLD consists of have and not have parts for human beings. For thousands years, human being has never stopped his steps for exploring its behaviors of all kinds. We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multispace came into being by purely logic.
Fast Backtracking Principles Applied to Find New Cages
 Ninth Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 1998
"... We describe how standard backtracking rules of thumb were successfully applied to the problem of characterizing (3; g) cages, the minimum order 3regular graphs of girth g. It took just 5 days of cpu time (compared to 259 days for previous authors) to verify the (3; 9)cages, and we were able to c ..."
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Cited by 9 (3 self)
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We describe how standard backtracking rules of thumb were successfully applied to the problem of characterizing (3; g) cages, the minimum order 3regular graphs of girth g. It took just 5 days of cpu time (compared to 259 days for previous authors) to verify the (3; 9)cages, and we were able to confirm that (3; 11)cages have order 112 for the first time ever. The lower bound for a (3; 13)cage is improved from 196 to 202 using the same approach. Also, we determined that a (3; 14)cage has order at least 258. 1 Cages In this paper, we consider finite undirected graphs. Any undefined notation follows Bondy and Murty [7]. The girth of a graph is the size of a smallest cycle. A (r; g) cage is an rregular graph of minimum order with girth g. It is known that (r; g)cages always exist [11]. Some nice pictures of small cages are given in [9, pp. 5458]. The classification of the cages has attracted much interest amongst the graph theory community, and many of these have special nam...
An Heuristic For Graph Symmetry Detection
 Proc. of Graph Drawing 99, Lecture Notes in Computer Science 1731:276285
, 1999
"... . We give a short introduction to an heuristic to find automorphisms in a graph such as axial, central or rotational symmetries. Using technics of factorial analysis, we embed the graph in an Euclidean space and try to detect and interpret the geometric symmetries of of the embedded graph. 1. Introd ..."
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Cited by 6 (1 self)
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. We give a short introduction to an heuristic to find automorphisms in a graph such as axial, central or rotational symmetries. Using technics of factorial analysis, we embed the graph in an Euclidean space and try to detect and interpret the geometric symmetries of of the embedded graph. 1. Introduction Testing whether a graph has any axial (rotational, central, respectively) symmetry is a NPcomplete problem [9]. Some restrictions (central symmetry with exactly one fixed vertex and no fixed edge) are polynomialy equivalent to the graph isomorphism test. Notice that this latter problem is not known to be either polynomial or NPcomplete in general. But several heuristics are known (e.g. [3]) and several restrictions leads to efficient algorithms: linear time isomorphism test for planar graphs [6] and interval graphs [8], polynomial time isomorphism test for fixed genus [10, 5], kcontractible graphs [12] and pairwise kseparable graphs [11], linear axial symmetry detection for plana...
Graphical Construction of Cubic Cages
 Congr. Numer
, 1995
"... In the first part of this paper, we present a graphical construction that connects most of the known cubic cages. The construction suggests a technique for searching for new cages which we have used in seeking the 9cage. In the second part, we describe the known cubic cages in another way, present ..."
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Cited by 3 (1 self)
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In the first part of this paper, we present a graphical construction that connects most of the known cubic cages. The construction suggests a technique for searching for new cages which we have used in seeking the 9cage. In the second part, we describe the known cubic cages in another way, present a highly symmetric 58vertex cubic graph with girth 9, and finally show how that graph fits into the partial order. 1 Definitions and Notation We will assume the usual machinery of graph theory. For a graph G; V (G) denotes the vertex set of G and E(G) denotes the edge set. The graphs we will consider will be simple, that is, they will not have loops (i.e., if e = fv 1 ; v 2 g 2 E(G) then v 1 6= v 2 ), nor will they have multiple edges. A cubic graph is simply a 3regular graph: every vertex has degree three. We define a cubic tree to be a connected tree in which all vertices are either of degree three or of degree one. We will have particular use for a certain sequence of cubic trees t 0 ;...
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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Cited by 1 (0 self)
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
Smarandache multispace theory (IV)  Applications to theoretical physics
, 2006
"... A Smarandache multispace is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various ..."
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A Smarandache multispace is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multispaces including three parts altogether. The first part is on algebraic multispaces with structures, such as those of multigroups, multirings, multivector spaces, multimetric spaces, multioperation systems and multimanifolds, also multivoltage graphs, multiembedding of a graph in an nmanifold, · · ·, etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudoplane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multispaces to theoretical physics, including the relativity theory, the Mtheory and the cosmology. Multispace models for pbranes and cosmos are constructed and some questions in cosmology are clarified by multispaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers.