Network theory and its associated techniques has tremendous impact in various discipline and research, from computer, engineering, architecture, humanities, social science to system biology. However in recent years epidemiology can be said to utilizes these potentials of network theory more than any other discipline. Graph which has been considered as the processor in network theory has a close relationship with epidemiology that dated as far back as early 1900 [1]. This is because the earliest models of infectious disease transfer were in a form of compartment which defines a graph even though adequate knowledge of mathematical computation and mechanistic behavior is scarce. This paper introduces a new type of disease propagation on network utilizing the potentials of neutrosophic algebraic group structures and graph theory.
{"title":"The Spread of Infectious Disease on Network Using Neutrosophic Algebraic Structure","authors":"A. Zubairu, A. A. Ibrahim","doi":"10.4236/OJDM.2017.72009","DOIUrl":"https://doi.org/10.4236/OJDM.2017.72009","url":null,"abstract":"Network theory and its associated techniques has tremendous impact in various discipline and research, from computer, engineering, architecture, humanities, social science to system biology. However in recent years epidemiology can be said to utilizes these potentials of network theory more than any other discipline. Graph which has been considered as the processor in network theory has a close relationship with epidemiology that dated as far back as early 1900 [1]. This is because the earliest models of infectious disease transfer were in a form of compartment which defines a graph even though adequate knowledge of mathematical computation and mechanistic behavior is scarce. This paper introduces a new type of disease propagation on network utilizing the potentials of neutrosophic algebraic group structures and graph theory.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"07 1","pages":"77-86"},"PeriodicalIF":0.0,"publicationDate":"2017-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45523966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a characterization of the boundaries of smooth strictly convex sets in the Euclidean plane R2 based on the existence and uniqueness of inscribed triangles.
基于内接三角形的存在唯一性，给出了欧氏平面R2上光滑严格凸集边界的一个刻画。
{"title":"Boundaries of Smooth Strictly Convex Sets in the Euclidean Plane R 2","authors":"H. Kramer","doi":"10.4236/OJDM.2017.72008","DOIUrl":"https://doi.org/10.4236/OJDM.2017.72008","url":null,"abstract":"We give a characterization of the boundaries of smooth strictly convex sets in the Euclidean plane R2 based on the existence and uniqueness of inscribed triangles.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"07 1","pages":"71-76"},"PeriodicalIF":0.0,"publicationDate":"2017-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44104306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yongqiang Zhao, Zhiming Fang, Yonggang Cui, Guoyan Ye, Zhijun Cao
It is hard to compute the competition number for a graph in general and characterizing a graph by its competition number has been one of important research problems in the study of competition graphs. Sano pointed out that it would be interesting to compute the competition numbers of some triangulations of a sphere as he got the exact value of the competition numbers of regular polyhedra. In this paper, we study the competition numbers of several kinds of triangulations of a sphere, and get the exact values of the competition numbers of a 24-hedron obtained from a hexahedron by adding a vertex in each face of the hexahedron and joining the vertex added in a face with the four vertices of the face, a class of dodecahedra constructed from a hexahedron by adding a diagonal in each face of the hexahedron, and a triangulation of a sphere with 3n (n≥2) vertices.
{"title":"Competition Numbers of Several Kinds of Triangulations of a Sphere","authors":"Yongqiang Zhao, Zhiming Fang, Yonggang Cui, Guoyan Ye, Zhijun Cao","doi":"10.4236/OJDM.2017.72006","DOIUrl":"https://doi.org/10.4236/OJDM.2017.72006","url":null,"abstract":"It is hard to compute the competition number for a graph in general and characterizing a graph by its competition number has been one of important research problems in the study of competition graphs. Sano pointed out that it would be interesting to compute the competition numbers of some triangulations of a sphere as he got the exact value of the competition numbers of regular polyhedra. In this paper, we study the competition numbers of several kinds of triangulations of a sphere, and get the exact values of the competition numbers of a 24-hedron obtained from a hexahedron by adding a vertex in each face of the hexahedron and joining the vertex added in a face with the four vertices of the face, a class of dodecahedra constructed from a hexahedron by adding a diagonal in each face of the hexahedron, and a triangulation of a sphere with 3n (n≥2) vertices.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"07 1","pages":"54-64"},"PeriodicalIF":0.0,"publicationDate":"2017-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49649778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tilings of p-groups are closely associated with error-correcting codes. In [1], M. Dinitz, attempting to generalize full-rank tilings of Zn2 to arbitrary finite abelian groups, was able to show that if p ≥5, then Znp admits full-rank tiling and left the case p=3, as an open question. The result proved in this paper the settles of the question for the case p=3.
{"title":"Non-Full Rank Factorization of Finite Abelian Groups","authors":"K. Amin","doi":"10.4236/OJDM.2017.72005","DOIUrl":"https://doi.org/10.4236/OJDM.2017.72005","url":null,"abstract":"Tilings of p-groups are closely associated with error-correcting codes. In [1], M. Dinitz, attempting to generalize full-rank tilings of Zn2 to arbitrary finite abelian groups, was able to show that if p ≥5, then Znp admits full-rank tiling and left the case p=3, as an open question. The result proved in this paper the settles of the question for the case p=3.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"07 1","pages":"720-726"},"PeriodicalIF":0.0,"publicationDate":"2017-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48403094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the least number of cells (the gap number) one needs to take out from a rectangle with integer sides of length at least 2 in order to be tiled by ribbon right trominoes is less than or equal to 4. If the sides of the rectangle are of length at least 5, then the gap number is less than or equal to 3. We also show that for the family of rectangles that have nontrivial minimal number of gaps, with probability 1, the only obstructions to tiling appear from coloring invariants. This is in contrast to what happens for simply connected regions. For that class of regions Conway and Lagarias found a tiling invariant that does not follow from coloring.
{"title":"Tiling Rectangles with Gaps by Ribbon Right Trominoes","authors":"P. Junius, V. Nitica","doi":"10.4236/OJDM.2017.72010","DOIUrl":"https://doi.org/10.4236/OJDM.2017.72010","url":null,"abstract":"We show that the least number of cells (the gap number) one needs to take out from a rectangle with integer sides of length at least 2 in order to be tiled by ribbon right trominoes is less than or equal to 4. If the sides of the rectangle are of length at least 5, then the gap number is less than or equal to 3. We also show that for the family of rectangles that have nontrivial minimal number of gaps, with probability 1, the only obstructions to tiling appear from coloring invariants. This is in contrast to what happens for simply connected regions. For that class of regions Conway and Lagarias found a tiling invariant that does not follow from coloring.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"07 1","pages":"87-102"},"PeriodicalIF":0.0,"publicationDate":"2017-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42752757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].
{"title":"On the 2-Domination Number of Complete Grid Graphs","authors":"Ramy S. Shaheen, Suhail Mahfud, Khames Almanea","doi":"10.4236/OJDM.2017.71004","DOIUrl":"https://doi.org/10.4236/OJDM.2017.71004","url":null,"abstract":"A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"07 1","pages":"32-50"},"PeriodicalIF":0.0,"publicationDate":"2017-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45735581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
With the challenge of quantum computing ahead, an analysis of number and representation adequate to the task is needed. Some clarifications on the combinatorial nature of representation are presented here; this is related to the foundations of digital representations of integers, and is thus also of interest in clarifying what numbers are and how they are used in pure and applied mathematics. The author hopes this work will help mathematicians and computer scientists better understand the nature of the Generalized Knapsack Code, a lattice-based code which the author believes to be particularly promising, and the use of number in computing in general.
{"title":"Number in Mathematical Cryptography","authors":"Nathan Hamlin","doi":"10.4236/OJDM.2017.71003","DOIUrl":"https://doi.org/10.4236/OJDM.2017.71003","url":null,"abstract":"With the challenge of quantum computing ahead, an analysis of number and representation adequate to the task is needed. Some clarifications on the combinatorial nature of representation are presented here; this is related to the foundations of digital representations of integers, and is thus also of interest in clarifying what numbers are and how they are used in pure and applied mathematics. The author hopes this work will help mathematicians and computer scientists better understand the nature of the Generalized Knapsack Code, a lattice-based code which the author believes to be particularly promising, and the use of number in computing in general.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"07 1","pages":"13-31"},"PeriodicalIF":0.0,"publicationDate":"2017-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48652924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider tilings of deficient rectangles by the set T4 of ribbon L-tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a 2×4 rectangle. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The crack divides the square in two parts of equal area. The number of tilings of a (4m+1)×(4m+1) deficient square by T4 is equal to the number of tilings by dominoes of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T4 is twice the number of tilings by dominoes of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. In both cases the counting is realized by an explicit function which is a bijection in the first case and a double cover in the second. If an extra 2×2 tile is added to T4 , we call the new tile set T+4. A tiling of a deficient rectangle by T+4 exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are either paired tetrominoes and each pair tiles a 2×4 rectangle, or are 2×2 squares. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The number of tilings of a (4m+1)×(4m+1) deficient square by T+4 is greater than the number of tilings by dominoes and monomers of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T+4 is greater than twice the number of tilings by dominoes and monomers of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. We also consider tilings by T4 and T+4 of other significant deficient regions. In particular we show that a deficient first quadrant, a deficient half strip, a deficient strip or a deficient bent strip cannot be tiled by T+4. Therefore T4 and T+4 give examples of tile sets that tile deficient rectangles but do not tile any deficient first quadrant, any deficient half strip, any deficient bent strip or any deficient strip.
{"title":"The Tilings of Deficient Squares by Ribbon L -Tetrominoes Are Diagonally Cracked","authors":"V. Nitica","doi":"10.4236/OJDM.2017.73015","DOIUrl":"https://doi.org/10.4236/OJDM.2017.73015","url":null,"abstract":"We consider tilings of deficient rectangles by the set T4 of ribbon L-tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a 2×4 rectangle. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The crack divides the square in two parts of equal area. The number of tilings of a (4m+1)×(4m+1) deficient square by T4 is equal to the number of tilings by dominoes of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T4 is twice the number of tilings by dominoes of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. In both cases the counting is realized by an explicit function which is a bijection in the first case and a double cover in the second. If an extra 2×2 tile is added to T4 , we call the new tile set T+4. A tiling of a deficient rectangle by T+4 exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are either paired tetrominoes and each pair tiles a 2×4 rectangle, or are 2×2 squares. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The number of tilings of a (4m+1)×(4m+1) deficient square by T+4 is greater than the number of tilings by dominoes and monomers of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T+4 is greater than twice the number of tilings by dominoes and monomers of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. We also consider tilings by T4 and T+4 of other significant deficient regions. In particular we show that a deficient first quadrant, a deficient half strip, a deficient strip or a deficient bent strip cannot be tiled by T+4. Therefore T4 and T+4 give examples of tile sets that tile deficient rectangles but do not tile any deficient first quadrant, any deficient half strip, any deficient bent strip or any deficient strip.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"07 1","pages":"165-176"},"PeriodicalIF":0.0,"publicationDate":"2017-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45132734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zhijun Cao, Yonggang Cui, Guoyan Ye, Yongqiang Zhao
For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and chara-cterizing a graph by its competition number has been one of important research problems in the study of competition graphs. A 2-connected planar graph G with minimum degree at least 3 is a pseudo-Halin graph if deleting the edges on the boundary of a single face f0 yields a tree. It is a Halin graph if the vertices of f0 all have degree 3 in G. In this paper, we compute the competition numbers of a kind of pseudo-Halin graphs.
{"title":"Competition Numbers of a Kind of Pseudo-Halin Graphs","authors":"Zhijun Cao, Yonggang Cui, Guoyan Ye, Yongqiang Zhao","doi":"10.4236/OJDM.2017.71002","DOIUrl":"https://doi.org/10.4236/OJDM.2017.71002","url":null,"abstract":"For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and chara-cterizing a graph by its competition number has been one of important research problems in the study of competition graphs. A 2-connected planar graph G with minimum degree at least 3 is a pseudo-Halin graph if deleting the edges on the boundary of a single face f0 yields a tree. It is a Halin graph if the vertices of f0 all have degree 3 in G. In this paper, we compute the competition numbers of a kind of pseudo-Halin graphs.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"07 1","pages":"3-12"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A familiar and natural decomposition of square matrices leads to the construction of a Pfaffian with the same value as the determinant of the square matrix.
一个熟悉而自然的方阵分解可以构造一个与方阵行列式值相同的普氏矩阵。
{"title":"A Note on a Natural Correspondence of a Determinant and Pfaffian","authors":"G. Miller","doi":"10.4236/OJDM.2016.71001","DOIUrl":"https://doi.org/10.4236/OJDM.2016.71001","url":null,"abstract":"A familiar and natural decomposition of square matrices leads to the construction of a Pfaffian with the same value as the determinant of the square matrix.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"07 1","pages":"1-2"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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