University of IsfahanTransactions on Combinatorics2251-865713420241201Methods for counting the intersections of slopes in the flat torus3053172777310.22108/toc.2023.135546.2023ENJohnBurkeDepartment of Mathematical Sciences,
Rhode Island College,
600 Mt. Pleasant Ave.
Providence, RI 02908MaitlandBurkeDepartment of Mathematical Sciences,
Rhode Island College,
600 Mt. Pleasant Ave.
Providence, RI 02908LeonardoPinheiroDepartment of Mathematical Sciences,
Rhode Island College,
600 Mt. Pleasant Ave.
Providence, RI 02908CameronRicherDepartment of Mathematical Sciences,
Rhode Island College,
600 Mt. Pleasant Ave.
Providence, RI 02908Journal Article20221108We define slopes in the flat torus as the set of equivalence classes of the solutions of linear equations in $\mathbb{R}^2$. The definition is equivalent to that of closed geodesics in the flat torus passing through the equivalence class of the point $(0,0)$. In this paper we derive formulas for counting the number of points in the intersection of multiple slopes in the flat torus.https://toc.ui.ac.ir/article_27773_5c824a257fe0d279edecdab4f38dbfe4.pdfUniversity of IsfahanTransactions on Combinatorics2251-865713420241201On the infinitary van der Waerden's Theorem3193252770510.22108/toc.2023.136693.2043ENShahramMohsenipourSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, IranJournal Article20230201We give a purely combinatorial proof for the infinitary van der Waerden's theorem.https://toc.ui.ac.ir/article_27705_6a1fcf61ec75ccb4887efdd9a9c4ed3d.pdfUniversity of IsfahanTransactions on Combinatorics2251-865713420241201A closed formula for the number of inequivalent ordered integer quadrilaterals with fixed perimeter3273342771010.22108/toc.2023.136913.2045ENBouroubiSadekFaculty of Mathematics, University of Sciences and Technology Houari Boumediene, P.B. 32 El-Alia, 16111, Bab Ezzouar Algiers, Algeria0000-0002-0691-6189Journal Article20230221Given an integer $n\geq4$, how many inequivalent quadrilaterals with ordered integer sides and perimeter $n$ are there? Denoting such number by $Q(n)$, the answer is given by the following closed formula:<br />\[<br />Q(n)=\left\{ \dfrac{1}{576}n\left( n+3\right) \left( 2n+3\right) -\dfrac{\left( -1\right) ^{n}}{192}n\left( n-5\right) \right\} \cdot<br />\]https://toc.ui.ac.ir/article_27710_0ca238274568bdcd23686f2944f7e43b.pdfUniversity of IsfahanTransactions on Combinatorics2251-865713420241201The inverse 1-median problem on a tree with transferring the weight of vertices3353502776710.22108/toc.2023.136964.2047ENTahereSayarFaculty of Mathematical Sciences, Shahrood University of Technology, University Blvd., Shahrood, Iran.JafarFathaliFaculty of Mathematical Sciences, Shahrood University of Technology, University Blvd., Shahrood, Iran0000-0003-1397-8529MojtabaGhiyasiDepartment of Management and Accounting, Faculty of Industrial Engineering and Management Sciences,
Shahrood University of Technology, Shahrood, Iran.Journal Article20230225In this paper, we investigate a case of the inverse 1-median problem on a tree by transferring the weights of vertices which has not been raised so far. This problem considers modifying the weights of vertices via transferring weights of the vertices with the minimum cost such that a given vertex of the tree becomes the 1-median with respect to the new weights. A linear programming model is proposed for this problem. The applicability and efficiency of the presented model are shown in numerical examples and a real-life problem dealing with transferring users in a social network.https://toc.ui.ac.ir/article_27767_8be004d2de59d9bb64f37acecec2f724.pdfUniversity of IsfahanTransactions on Combinatorics2251-865713420241201On topological charge indices of graphs3513612751710.22108/toc.2023.137285.2062ENMohammadShahryariDepartment of Mathematics, College of Science, Sultan Qaboos University, Muscat, Oman0000-0002-0463-2182GholamrezaVakili-NezhaadDepartment of Petroleum and Chemical Engineering, College of Engineering, Sultan Qaboos University, Muscat, Oman0000-0002-3011-678XJournal Article20230411We introduce a fast method of computing the topological charge indices of simple graphs (molecules) which does not require matrices of large sizes. For the case of trees, we give a compact formula and in the general case we obtain upper and lower bounds for the charge indices. We give concrete examples of trees and molecules with their charge indices computed using our method.https://toc.ui.ac.ir/article_27517_8d6b0407a642e3f3320021d2f198c115.pdfUniversity of IsfahanTransactions on Combinatorics2251-865713420241201On the $sd_{b}$-critical graphs3633752788710.22108/toc.2023.137558.2069ENMohamedZamimeDepartment of Technology, University Yahia Fares of Medea, c.p 26000, Medea, AlgeriaJournal Article20230502A $b$-coloring of a graph\ $G$ is a proper coloring of its vertices such that each color class contains a vertex that has a neighbor in every other color classes. The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the largest integer $k$ such that $G$ admits a $b$-coloring with $k$ colors. Let $G_{e}$ be the graph obtained from $G$ by subdividing the edge $e $. A graph $G$ is $sd_{b}$-critical if $b(G_{e})<b(G)$ holds for any edge $e$ of $G$. In this paper, we first present \ several basic properties of $sd_{b} $-critical graphs and then we give a characterization of $sd_{b}$-critical $P_{4}$-sparse graphs and $sd_{b}$-critical quasi-line graphs.https://toc.ui.ac.ir/article_27887_75f64213393ed8de86857b9577227fcd.pdfUniversity of IsfahanTransactions on Combinatorics2251-865713420241201An existence theorem of perfect matching on $k$-partite $k$-uniform hypergraphs via distance spectral radius3773852794510.22108/toc.2023.137937.2077ENLeiZhangDepartment of Mathematics and Statistics, Qinghai Normal University, Xining, ChinaHaizhenRenDepartment of Mathematics and Statistics, Qinghai Normal University, Xining, China0000-0001-5609-5924Journal Article20230606Let $n_1, n_2,\ldots,n_k$ be integers and $V_1, V_2,\ldots,V_k$ be disjoint vertex sets with $|V_i|=n_i$ for each $i= 1, 2,\ldots,k$. A $k$-partite $k$-uniform hypergraph on vertex classes $V_1, V_2,\ldots,V_k$ is defined to be the $k$-uniform hypergraph whose edge set consists of the $k$-element subsets $S$ of $V_1 \cup V_2 \cup \cdots \cup V_k$ such that $|S\cap V_i|=1$ for all $i= 1, 2,\ldots,k$. We say that it is balanced if $n_1=n_2=\cdots=n_k$. In this paper, we give a distance spectral radius condition to guarantee the existence of perfect matching in $k$-partite $k$-uniform hypergraphs, this result generalize the result of Zhang and Lin [Perfect matching and distance spectral radius in graphs and bipartite graphs, <em>Discrete Appl. Math.,</em> <strong>304 </strong>(2021) 315-322].https://toc.ui.ac.ir/article_27945_1541186ba65820112fd9979a333aaf05.pdf