Pub Date : 2021-02-22DOI: 10.26493/2590-9770.1404.61e
E. Sotnikova, A. Valyuzhenich
In this work we present a survey of results on the problem of finding the minimum cardinality of the support of eigenfunctions of graphs.
本文对图的特征函数支持的最小基数问题的结果进行了综述。
{"title":"Minimum supports of eigenfunctions of graphs: a survey","authors":"E. Sotnikova, A. Valyuzhenich","doi":"10.26493/2590-9770.1404.61e","DOIUrl":"https://doi.org/10.26493/2590-9770.1404.61e","url":null,"abstract":"In this work we present a survey of results on the problem of finding the minimum cardinality of the support of eigenfunctions of graphs.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43911828","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}
Pub Date : 2020-09-22DOI: 10.26493/2590-9770.1356.D19
M. Changat, Prasanth G. Narasimha-Shenoi, Ferdoos Hossein Nezhad, M. Kovse, S. Mohandas, Abisha Ramachandran, P. Stadler
Genetic Algorithms typically invoke crossover operators to two parents. The transit set R k ( x, y ) comprises all offsprings of this form. It forms the tope set of an uniform oriented matroid with Vapnik-Chervonenkis dimension k + 1 . The Topological Representation Theorem for oriented matroids thus implies a representation in terms of pseudosphere arrangements. This makes it possible to study 2 -point crossover in detail and to characterize the partial cubes defined by the transit sets of two-point crossover.
{"title":"Transit sets of two-point crossover","authors":"M. Changat, Prasanth G. Narasimha-Shenoi, Ferdoos Hossein Nezhad, M. Kovse, S. Mohandas, Abisha Ramachandran, P. Stadler","doi":"10.26493/2590-9770.1356.D19","DOIUrl":"https://doi.org/10.26493/2590-9770.1356.D19","url":null,"abstract":"Genetic Algorithms typically invoke crossover operators to two parents. The transit set R k ( x, y ) comprises all offsprings of this form. It forms the tope set of an uniform oriented matroid with Vapnik-Chervonenkis dimension k + 1 . The Topological Representation Theorem for oriented matroids thus implies a representation in terms of pseudosphere arrangements. This makes it possible to study 2 -point crossover in detail and to characterize the partial cubes defined by the transit sets of two-point crossover.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42156357","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}
Pub Date : 2020-09-08DOI: 10.26493/2590-9770.1381.332
M. Conder, Isabel Holm, T. Tucker
{"title":"Observations and answers to questions about edge-transitive maps","authors":"M. Conder, Isabel Holm, T. Tucker","doi":"10.26493/2590-9770.1381.332","DOIUrl":"https://doi.org/10.26493/2590-9770.1381.332","url":null,"abstract":"","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41772048","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}
Pub Date : 2020-01-29DOI: 10.26493/2590-9770.1354.B40
Antonio Montero, A. I. Weiss
We show that every non-degenerate regular polytope can be used to construct a thin, residually-connected, chamber-transitive incidence geometry, i.e. a regular hypertope, with a tail-triangle Coxeter diagram. We discuss several interesting examples derived when this construction is applied to generalised cubes. In particular, we produce an example of a rank $5$ finite locally spherical proper hypertope of hyperbolic type.
{"title":"Locally spherical hypertopes from generalised cubes","authors":"Antonio Montero, A. I. Weiss","doi":"10.26493/2590-9770.1354.B40","DOIUrl":"https://doi.org/10.26493/2590-9770.1354.B40","url":null,"abstract":"We show that every non-degenerate regular polytope can be used to construct a thin, residually-connected, chamber-transitive incidence geometry, i.e. a regular hypertope, with a tail-triangle Coxeter diagram. We discuss several interesting examples derived when this construction is applied to generalised cubes. In particular, we produce an example of a rank $5$ finite locally spherical proper hypertope of hyperbolic type.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43046556","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}
Pub Date : 2019-12-17DOI: 10.26493/2590-9770.1353.C0E
K. Kostousov
For a positive integer $d$, a connected graph $Gamma$ is a symmetrical 2-extension of the $d$-dimensional grid $Lambda^d$ if there exists a vertex-tran-sitive group $G$ of automorphisms of $Gamma$ and its imprimitivity system $sigma$ with blocks of order 2 such that there exists an isomorphism $varphi$ of the quotient graph $Gamma/sigma$ onto $Lambda^d$. The tuple $(Gamma, G, sigma, varphi)$ with specified components is called a realization of the symmetrical 2-extension $Gamma$ of the grid $Lambda^{d}$. Two realizations $(Gamma_1, G_1,$ $sigma_1, varphi_1)$ and $(Gamma_2, G_2, sigma_2, varphi_2)$ are called equivalent if there exists an isomorphism of the graph $Gamma_1$ onto $Gamma_2$ which maps $sigma_1$ onto $sigma_2$. V. Trofimov proved that, up to equivalence, there are only finitely many realizations of symmetrical $2$-extensions of $Lambda^{d}$ for each positive integer $d$. E. Konovalchik and K. Kostousov found all, up to equivalence, realizations of symmetrical 2-extensions of the grid $Lambda^2$. In this work we found all, up to equivalence, realizations $(Gamma, G, sigma, varphi)$ of symmetrical 2-extensions of the grid $Lambda^3$ for which only the trivial automorphism of $Gamma$ preserves all blocks of $sigma$ (we prove that there are 5573 such realizations, and that among corresponding graphs $Gamma$ there are 5350 pairwise non-isomorphic).
{"title":"Symmetrical 2-extensions of the 3-dimensional grid. I","authors":"K. Kostousov","doi":"10.26493/2590-9770.1353.C0E","DOIUrl":"https://doi.org/10.26493/2590-9770.1353.C0E","url":null,"abstract":"For a positive integer $d$, a connected graph $Gamma$ is a symmetrical 2-extension of the $d$-dimensional grid $Lambda^d$ if there exists a vertex-tran-sitive group $G$ of automorphisms of $Gamma$ and its imprimitivity system $sigma$ with blocks of order 2 such that there exists an isomorphism $varphi$ of the quotient graph $Gamma/sigma$ onto $Lambda^d$. The tuple $(Gamma, G, sigma, varphi)$ with specified components is called a realization of the symmetrical 2-extension $Gamma$ of the grid $Lambda^{d}$. Two realizations $(Gamma_1, G_1,$ $sigma_1, varphi_1)$ and $(Gamma_2, G_2, sigma_2, varphi_2)$ are called equivalent if there exists an isomorphism of the graph $Gamma_1$ onto $Gamma_2$ which maps $sigma_1$ onto $sigma_2$. V. Trofimov proved that, up to equivalence, there are only finitely many realizations of symmetrical $2$-extensions of $Lambda^{d}$ for each positive integer $d$. E. Konovalchik and K. Kostousov found all, up to equivalence, realizations of symmetrical 2-extensions of the grid $Lambda^2$. In this work we found all, up to equivalence, realizations $(Gamma, G, sigma, varphi)$ of symmetrical 2-extensions of the grid $Lambda^3$ for which only the trivial automorphism of $Gamma$ preserves all blocks of $sigma$ (we prove that there are 5573 such realizations, and that among corresponding graphs $Gamma$ there are 5350 pairwise non-isomorphic).","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45534970","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}
Pub Date : 2019-12-05DOI: 10.26493/2590-9770.1365.884
G. Jones
Infinite analogues of the Paley graphs are constructed, based on uncountably many infinite but locally finite fields. Weil's estimate for character sums shows that they are all isomorphic to the random or universal graph of ErdH os, Renyi and Rado. Automorphism groups and connections with model theory are considered.
{"title":"Infinite Paley graphs","authors":"G. Jones","doi":"10.26493/2590-9770.1365.884","DOIUrl":"https://doi.org/10.26493/2590-9770.1365.884","url":null,"abstract":"Infinite analogues of the Paley graphs are constructed, based on uncountably many infinite but locally finite fields. Weil's estimate for character sums shows that they are all isomorphic to the random or universal graph of ErdH os, Renyi and Rado. Automorphism groups and connections with model theory are considered.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49342335","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}
Pub Date : 2019-11-21DOI: 10.26493/2590-9770.1338.0b2
Pablo Spiga
In this paper we show that two distinct conjectures, the first proposed by Babai and Godsil in $1982$ and the second proposed by Xu in $1998$, concerning the asymptotic enumeration of Cayley graphs are in fact equivalent. This result follows from a more general theorem concerning the asymptotic enumeration of a certain family of Cayley graphs.
{"title":"On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu concerning the enumeration of Cayley graphs","authors":"Pablo Spiga","doi":"10.26493/2590-9770.1338.0b2","DOIUrl":"https://doi.org/10.26493/2590-9770.1338.0b2","url":null,"abstract":"In this paper we show that two distinct conjectures, the first proposed by Babai and Godsil in $1982$ and the second proposed by Xu in $1998$, concerning the asymptotic enumeration of Cayley graphs are in fact equivalent. This result follows from a more general theorem concerning the asymptotic enumeration of a certain family of Cayley graphs.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44492684","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}
Pub Date : 2019-10-28DOI: 10.26493/2590-9770.1624.a3d
S. Mirafzal
Let $G$ be a finite abelian group written additively with identity $0$, and $Omega$ be an inverse closed generating subset of $G$ such that $0notin Omega$. We say that $ Omega $ has the property lqlq{}$us$rqrq{} (unique summation), whenever for every $0 neq gin G$ if there are $s_1,s_2,s_3, s_4 in Omega $ such that $s_1+s_2=g=s_3+s_4 $, then we have ${s_1,s_2 } = {s_3,s_4 }$. We say that a Cayley graph $Gamma=Cay(G;Omega)$ is a $us$-$Cayley graph$, whenever $G$ is an abelian group and the generating subset $Omega$ has the property lqlq{}$us$rqrq{}. In this paper, we show that if $Gamma=Cay(G;Omega)$ is a $us$-$Cayley graph$, then $Aut(Gamma)=L(G)rtimes A$, where $L(G)$ is the left regular representation of $G$ and $A$ is the group of all automorphism groups $theta$ of the group $G$ such that $theta(Omega)=Omega$. Then, as some applications, we explicitly determine the automorphism groups of some classes of graphs including M"{o}bius ladders and $k$-ary $n$-cubes.
{"title":"On the automorphism groups of us-Cayley graphs","authors":"S. Mirafzal","doi":"10.26493/2590-9770.1624.a3d","DOIUrl":"https://doi.org/10.26493/2590-9770.1624.a3d","url":null,"abstract":"Let $G$ be a finite abelian group written additively with identity $0$, and $Omega$ be an inverse closed generating subset of $G$ such that $0notin Omega$. We say that $ Omega $ has the property lqlq{}$us$rqrq{} (unique summation), whenever for every $0 neq gin G$ if there are $s_1,s_2,s_3, s_4 in Omega $ such that $s_1+s_2=g=s_3+s_4 $, then we have ${s_1,s_2 } = {s_3,s_4 }$. We say that a Cayley graph $Gamma=Cay(G;Omega)$ is a $us$-$Cayley graph$, whenever $G$ is an abelian group and the generating subset $Omega$ has the property lqlq{}$us$rqrq{}. In this paper, we show that if $Gamma=Cay(G;Omega)$ is a $us$-$Cayley graph$, then $Aut(Gamma)=L(G)rtimes A$, where $L(G)$ is the left regular representation of $G$ and $A$ is the group of all automorphism groups $theta$ of the group $G$ such that $theta(Omega)=Omega$. Then, as some applications, we explicitly determine the automorphism groups of some classes of graphs including M\"{o}bius ladders and $k$-ary $n$-cubes.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":"7 1","pages":"1"},"PeriodicalIF":0.0,"publicationDate":"2019-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69338986","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}
Pub Date : 2019-08-03DOI: 10.26493/2590-9770.1314.f6d
G. Jones
Building on earlier work of Biggs, James, Wilson and the author, and using the Graver-Watkins description of the 14 classes of edge-transitive maps, we complete the classification of the edge-transitive embeddings of complete graphs.
{"title":"Edge-transitive embeddings of complete graphs","authors":"G. Jones","doi":"10.26493/2590-9770.1314.f6d","DOIUrl":"https://doi.org/10.26493/2590-9770.1314.f6d","url":null,"abstract":"Building on earlier work of Biggs, James, Wilson and the author, and using the Graver-Watkins description of the 14 classes of edge-transitive maps, we complete the classification of the edge-transitive embeddings of complete graphs.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43687725","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}
{"title":"On strictly Deza graphs derived from the Berlekamp-van Lint-Seidel graph","authors":"S. Zaw","doi":"10.26493/2590-9770.1335.2FA","DOIUrl":"https://doi.org/10.26493/2590-9770.1335.2FA","url":null,"abstract":"In this paper, we find strictly Deza graphs that can be obtained from the Berlekamp-van Lint-Seidel graph by applying dual Seidel switching.","PeriodicalId":36246,"journal":{"name":"Art of Discrete and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49585850","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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