{"title":"On the $$A_{alpha }$$-Spectra of Some Join Graphs","authors":"M. Basunia, Iswar Mahato, M. Kannan","doi":"10.1007/s40840-021-01166-z","DOIUrl":"https://doi.org/10.1007/s40840-021-01166-z","url":null,"abstract":"","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89933670","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":"Antipalindromic numbers","authors":"L'ubomíra Dvoráková, Stanislav Kruml, David Ryzak","doi":"10.14311/ap.2021.61.0428","DOIUrl":"https://doi.org/10.14311/ap.2021.61.0428","url":null,"abstract":"Everybody has certainly heard about palindromes: words that stay the same when read backwards. For instance kayak, radar, or rotor. Mathematicians are interested in palindromic numbers: positive integers whose expansion in a certain integer base is a palindrome. The following problems are studied: palindromic primes, palindromic squares and higher powers, multibased palindromic numbers, etc. In this paper, we define and study antipalindromic numbers: positive integers whose expansion in a certain integer base is an antipalindrome. We present new results concerning divisibility and antipalindromic primes, antipalindromic squares and higher powers, and multibased antipalindromic numbers. We provide a user-friendly application for all studied questions.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"427 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76942650","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":"Combinatorics of Multicompositions","authors":"Brian Hopkins, S. Ouvry","doi":"10.1007/978-3-030-67996-5_16","DOIUrl":"https://doi.org/10.1007/978-3-030-67996-5_16","url":null,"abstract":"","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80094016","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":"Coloring the normalized Laplacian for oriented hypergraphs","authors":"A. Abiad, R. Mulas, Dong Zhang","doi":"10.1016/j.laa.2021.07.018","DOIUrl":"https://doi.org/10.1016/j.laa.2021.07.018","url":null,"abstract":"","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77287117","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":"Bordered Complex Hadamard Matrices and Strongly Regular Graphs","authors":"Takuya Ikuta, A. Munemasa","doi":"10.4036/IIS.2020.R.03","DOIUrl":"https://doi.org/10.4036/IIS.2020.R.03","url":null,"abstract":"We consider bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph. Examples include a Butson-type complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph due to J. Wallis, and a family of Hadamard matrices given by Singh and Dubey. In this paper, we show that there is also a non Butson-type complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph, and prove that there are no other bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75554026","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 the modular Jones polynomial","authors":"G. Pagel","doi":"10.5802/crmath.106","DOIUrl":"https://doi.org/10.5802/crmath.106","url":null,"abstract":"A major problem in knot theory is to decide whether the Jones polynomial detects the unknot. In this paper we study a weaker related problem, namely whether the Jones polynomial reduced modulo an integer $n$ detects the unknot. The answer is known to be negative for $n=2^k$ with $kgeq 1$ and $n=3$. Here we show that if the answer is negative for some $n$, then it is negative for $n^k$ with any $kgeq 1$. In particular, for any $kgeq 1$, we construct nontrivial knots whose Jones polynomial is trivial modulo~$3^k$.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85743152","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 a Zero-Sum Problem Arising From Factorization Theory","authors":"Aqsa Bashir, A. Geroldinger, Qinghai Zhong","doi":"10.1007/978-3-030-67996-5_2","DOIUrl":"https://doi.org/10.1007/978-3-030-67996-5_2","url":null,"abstract":"","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"115 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86032685","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 THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II","authors":"M. Abdulla, Ayman Badawi","doi":"10.24330/ieja.768135","DOIUrl":"https://doi.org/10.24330/ieja.768135","url":null,"abstract":"In 2015, the second-named author introduced the dot product graph associated to a commutative ring A. Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × · · · × A (n times). We recall that the total dot product graph of R is the (undirected) graph TD(R) with vertices R∗ = R {(0, 0, ...,0)}, and two distinct vertices x and y are adjacent if and only if x · y = 0 ∈ A (where x · y denotes the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of TD(R) with vertices Z(R) = Z(R){(0, 0, ..., 0)}. Let U(R) denote the set of all units of R. Then the unit dot product graph of R is the induced subgraph UD(R) of TD(R) with vertices U(R). In this paper, we study the structure of TD(R), UD(R), and ZD(R) when A = Zn or A = GF (pn), the finite field with pn elements, where n ≥ 2 and p is a prime positive integer. 1991 Mathematics Subject Classification Primary: 13A15; Secondary: 13B99; 05C99","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"150 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72687142","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":"Polynomials related to chromatic polynomials","authors":"F. Dong","doi":"10.1142/9789812569462_0011","DOIUrl":"https://doi.org/10.1142/9789812569462_0011","url":null,"abstract":"For a simple graph $G$, let $chi(G,x)$ denote the chromatic polynomial of $G$. This manuscript introduces some polynomials which are related to chromatic polynomial and their relations.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80221856","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":"Matchings in regular graphs: minimizing the partition function","authors":"M'arton Borb'enyi, P'eter Csikv'ari","doi":"10.22108/TOC.2020.123763.1742","DOIUrl":"https://doi.org/10.22108/TOC.2020.123763.1742","url":null,"abstract":"For a graph $G$ on $v(G)$ vertices let $m_k(G)$ denote the number of matchings of size $k$, and consider the partition function $M_{G}(lambda)=sum_{k=0}^nm_k(G)lambda^k$. In this paper we show that if $G$ is a $d$--regular graph and $0 frac{1}{v(K_{d+1})}ln M_{K_{d+1}}(lambda).$$ The same inequality holds true if $d=3$ and $lambda<0.3575$. More precise conjectures are also given.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86001626","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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