{"title":"Circulant almost cross intersecting families","authors":"Michal Parnas","doi":"10.26493/2590-9770.1414.F67","DOIUrl":"https://doi.org/10.26493/2590-9770.1414.F67","url":null,"abstract":"Let $mathcal{F}$ and $mathcal{G}$ be two $t$-uniform families of subsets over $[k] = {1,2,...,k}$, where $|mathcal{F}| = |mathcal{G}|$, and let $C$ be the adjacency matrix of the bipartite graph whose vertices are the subsets in $mathcal{F}$ and $mathcal{G}$, and there is an edge between $Ain mathcal{F}$ and $B in mathcal{G}$ if and only if $A cap B neq emptyset$. The pair $(mathcal{F},mathcal{G})$ is $q$-almost cross intersecting if every row and column of $C$ has exactly $q$ zeros. \u0000We consider $q$-almost cross intersecting pairs that have a circulant intersection matrix $C_{p,q}$, determined by a column vector with $p > 0$ ones followed by $q > 0$ zeros. This family of matrices includes the identity matrix in one extreme, and the adjacency matrix of the bipartite crown graph in the other extreme. \u0000We give constructions of pairs $(mathcal{F},mathcal{G})$ whose intersection matrix is $C_{p,q}$, for a wide range of values of the parameters $p$ and $q$, and in some cases also prove matching upper bounds. Specifically, we prove results for the following values of the parameters: (1) $1 leq p leq 2t-1$ and $1 leq q leq k-2t+1$. (2) $2t leq p leq t^2$ and any $q> 0$, where $k geq p+q$. (3) $p$ that is exponential in $t$, for large enough $k$. \u0000Using the first result we show that if $k geq 4t-3$ then $C_{2t-1,k-2t+1}$ is a maximal isolation submatrix of size $ktimes k$ in the $0,1$-matrix $A_{k,t}$, whose rows and columns are labeled by all subsets of size $t$ of $[k]$, and there is a one in the entry on row $x$ and column $y$ if and only if subsets $x,y$ intersect.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"140 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116518918","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}
M. Knor, J. Komorník, R. Škrekovski, Aleksandra Tepeh
{"title":"Some remarks on Balaban and sum-Balaban index","authors":"M. Knor, J. Komorník, R. Škrekovski, Aleksandra Tepeh","doi":"10.26493/2590-9770.1241.7f8","DOIUrl":"https://doi.org/10.26493/2590-9770.1241.7f8","url":null,"abstract":"In the paper we study maximal values of Balaban and sum-Balaban index, and correct some results appearing in the literature which are only partially correct. Henceforth, we were able to solve a conjecture of M. Aouchiche, G. Caporossi and P. Hansen regarding the comparison of Balaban and Randić index. In addition, we showed that for every k and large enough n, the first k graphs of order n with the largest value of Balaban index are trees. We conclude the paper with a result about the accumulation points of sum-Balaban index.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"120 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127757102","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":"Distributions of restricted rotation distances","authors":"S. Cleary, Haris Nadeem","doi":"10.26493/2590-9770.1374.BF0","DOIUrl":"https://doi.org/10.26493/2590-9770.1374.BF0","url":null,"abstract":"Rotation distances measure the differences in structure between rooted ordered binary trees. The one-dimensional skeleta of associahedra are rotation graphs, where two vertices representing trees are connected by an edge if they differ by a single rotation. There are no known efficient algorithms to compute rotation distance between trees and thus distances in rotation graphs. Limiting the allowed locations of where rotations are permitted gives rise to a number of notions of restricted rotation distances. Allowing rotations at a minimal such set of locations gives restricted rotation distance. There are linear-time algorithms to compute restricted rotation distance, where there are only two permitted locations for rotations to occur. The associated restricted rotation graph has an efficient distance algorithm. There are linear upper and lower bounds on restricted rotation distance with respect to the sizes of the reduced tree pairs. Here, we experimentally investigate the expected restricted rotation distance between two trees selected at random of increasing size and find that it lies typically in a narrow band well within the earlier proven linear upper and lower bounds.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124550714","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 median and quartile sets of ordered random variables","authors":"I. Banič, J. Žerovnik","doi":"10.26493/2590-9770.1326.9fd","DOIUrl":"https://doi.org/10.26493/2590-9770.1326.9fd","url":null,"abstract":"We give new results about the set of all medians, the set of all first quartiles and the set of all third quartiles of a finite dataset. We also give new and interesting results about relationships between these sets. We also use these results to provide an elementary correctness proof of the Langford's doubling method.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122010348","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":"Open problems from NCS 2018","authors":"Jonathan Leech, João Pita Costa","doi":"10.26493/2590-9770.1332.594","DOIUrl":"https://doi.org/10.26493/2590-9770.1332.594","url":null,"abstract":"All participants in the workshop NCS 2018 were invited to submit a list of open problems, typically problems relevant to the subject matter of their talk. These problems follow, grouped according the particular individual presenting them. These individuals appear with their problem sets in alphabetical order, the sole exception being that the Honoree of the workshop appears first. Some of the presenters give a bit of background to accompany their problems. In other cases, such as myself, the presenter assumes that enough background was given in their talk as published herein. Thus the reader is invited to refer back to the relevant article. It is hoped that in the months to follow some of these problems will be solved, or at least, considerable light will be shed on them.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126709584","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":"Dual binary discriminator varieties","authors":"R. Bignall, M. Spinks","doi":"10.26493/2590-9770.1324.5b2","DOIUrl":"https://doi.org/10.26493/2590-9770.1324.5b2","url":null,"abstract":"Left normal bands, strongly distributive skew lattices, implicative BCS-algebras, skew Boolean algebras, skew Boolean intersection algebras, and certain other non-commutative structures occur naturally as term reducts in the study of ternary discriminator algebras and the varieties that they generate, giving rise thereby to various classes of pointed discriminator varieties1 that generalise the class of pointed ternary discriminator varieties. For each such class of varieties there is a corresponding pointed discriminator function that generalises the ternary discriminator. In this paper some of the classes of pointed discriminator varieties that are contained in the class of dual binary discriminator varieties are characterised. A key unifying property is that the principal ideals of an algebra in a dual binary discriminator variety are entirely determined by the dual binary discriminator term for that variety.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130168670","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 of degree 4 over finite fields representing quadratic residues","authors":"Shao-Fei Du, Klavdija Kutnar, D. Marušič","doi":"10.26493/2590-9770.1330.993","DOIUrl":"https://doi.org/10.26493/2590-9770.1330.993","url":null,"abstract":"It is proved that in a finite field F of prime order p, where p is not one of finitely many exceptions, for every polynomial f(x) ∈ F [x] of degree 4 that has a nonzero constant term and is not of the form αg(x) there exists a primitive root β ∈ F such that f(β) is a quadratic residue in F . This refines a result of Madden and Vélez from 1982 about polynomials that represent quadratic residues at primitive roots.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124460756","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":"Infinite benzenoids","authors":"N. Bašić","doi":"10.26493/2590-9770.1228.eb5","DOIUrl":"https://doi.org/10.26493/2590-9770.1228.eb5","url":null,"abstract":"A family of benzenoids, called convex benzenoids, was introduced in 2012 by Cruz, Gutman and Rada. In a later paper by the present author et al., several equivalent characterisations of convex benzenoids have been given and their equivalence was proved. Along the way an infinite benzenoid called the half-plane was used for the purpose of theoretical reasoning. In this short paper, some properties of infinite benzenoids are discussed. It is proved that their boundary consists of countably many connected components. Convex infinite benzenoids are classified and it is proved that there are only countably many convex infinite benzenoids, whilst there are uncountably many infinite (non-convex) benzenoids. We also show that there are countably many infinite benzenoids which have a finite number of 1s in their boundary-edges code.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115185882","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":"Ω-lattices from skew lattices","authors":"J. Jovanović, A. Tepavčević","doi":"10.26493/2590-9770.1309.6c0","DOIUrl":"https://doi.org/10.26493/2590-9770.1309.6c0","url":null,"abstract":"Some notes on construction of Ω-lattices from special types of skew lattices, using lattices of weak congruences of skew lattices, are presented.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132915288","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}
Karin Cvetko-Vah, M. Kinyon, Jonathan Leech, Tomavz Pisanski
{"title":"Regular antilattices","authors":"Karin Cvetko-Vah, M. Kinyon, Jonathan Leech, Tomavz Pisanski","doi":"10.26493/2590-9770.1333.152","DOIUrl":"https://doi.org/10.26493/2590-9770.1333.152","url":null,"abstract":"Antilattices $(S;lor, land)$ for which the Green's equivalences $mathcal L_{(lor)}$, $mathcal R_{(lor)}$, $mathcal L_{(land)}$ and $mathcal R_{(land)}$ are all congruences of the entire antilattice are studied and enumerated.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"176 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132318708","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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