Transactions on Combinatorics
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Transactions on Combinatoricsendaily1Sun, 01 Dec 2024 00:00:00 +0330Sun, 01 Dec 2024 00:00:00 +0330Methods for counting the intersections of slopes in the flat torus
https://toc.ui.ac.ir/article_27773.html
We 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.On the infinitary van der Waerden's Theorem
https://toc.ui.ac.ir/article_27705.html
We give a purely combinatorial proof for the infinitary van der Waerden's theorem.A closed formula for the number of inequivalent ordered integer quadrilaterals with fixed perimeter
https://toc.ui.ac.ir/article_27710.html
Given 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:\[Q(n)=\left\{ \dfrac{1}{576}n\left( n+3\right) \left( 2n+3\right) -\dfrac{\left( -1\right) ^{n}}{192}n\left( n-5\right) \right\} \cdot\]The inverse 1-median problem on a tree with transferring the weight of vertices
https://toc.ui.ac.ir/article_27767.html
In 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.On topological charge indices of graphs
https://toc.ui.ac.ir/article_27517.html
We 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.On the $sd_{b}$-critical graphs
https://toc.ui.ac.ir/article_27887.html
A $b$-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 $b(G_{e})&lt;b(G)$ 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.An existence theorem of perfect matching on $k$-partite $k$-uniform hypergraphs via distance spectral radius
https://toc.ui.ac.ir/article_27945.html
Let $n_1, n_2,\ldots,n_k$ 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 &nbsp;[Perfect matching and distance spectral radius in graphs and bipartite graphs, Discrete Appl. Math., 304 (2021) 315-322].Line graphs of directed graphs I
https://toc.ui.ac.ir/article_28073.html
We determine the forbidden induced subgraphs for the intersection of the classes of chordal bipartite graphs and line graphs of acyclic directed graphs. This is a first step towards finding the forbidden induced subgraphs for the class of line graphs of directed graphs.Shield tilings
https://toc.ui.ac.ir/article_28096.html
We provide a complete description of the edge-to-edge tilings with a regular triangle and a shield-shaped hexagon with no right angle. The case of a hexagon with a right angle is also briefly discussed.Cayley hypergraph over polygroups
https://toc.ui.ac.ir/article_28119.html
Comer introduced a class of hypergroups, using the name of polygroups. He emphasized the importance of polygroups, by analyzing them in connections to graphs, relations, Boolean and cylindric algebras. Indeed, polygroups are multi valued systems that satisfy group like axioms. Given a polygroup with a finite generating set, we can form a Cayley hypergraph for that polygroup with respect to that generating set. This helps us to better understand and investigate polygroup structures. More precisely,in this paper, we introduce the construction of Cayley hypergraphs over polygroups, say $CH(\mathbf{P},S)$ such that $\mathbf{P}$ is a polygroup and $\langle S\rangle =P$. We investigate some properties of them. It is well known to give a constructing for building a big polygroup from two small ones. This structure is called extensionof polygroups. In particular, we describe the connection between Cayley hypergraphs over extension of two polygroups and Cartesian product of two Cayley hypergraphs.The degree-associated reconstruction number of an unicentroidal tree
https://toc.ui.ac.ir/article_28135.html
As we know, by deleting one vertex of a graph $G$, we have a subgraph of $G$ called a card of $G$. Also, investigation of that each graph with at least three vertices is determined by its multiset of cards, is called the reconstruction conjecture and the minimum number of dacards that determine $G$ is denoted the degree-associated reconstruction number $drn(G)$. Barrus and West conjectured that $drn(G) \leq 2$ for all but finitely many trees. A tree is unicentroidal or bicentroidal when it has one or two centroids, respectively. An unicentroidal tree $T$ with centroid $v$ is symmetrical if for two neighbours of $u$ and $u'$ of $v$, there exists an automorphism on $T$ mapping $u$ to $u'$. In \cite{Shad}, Shadravan and Borzooei proved that the conjecture is true for any non-symmetrical unicentroidal tree. In this paper, we proved that for any symmetrical unicentroidal tree $T$, $drn(T) \leq 2$. So, we concluded that the conjecture is true for any unicentroidal tree.Commutative rings introduce a class of identifiable graphs
https://toc.ui.ac.ir/article_28193.html
Let $R$ be a commutative ring with identity, and $ \mathrm{A}(R) $ be the set of ideals with non-zero annihilator. The annihilating-ideal graph of $ R $ is defined as the graph $AG(R)$ with the vertex set $ \mathrm{A}(R)^{*}=\mathrm{A}(R)\setminus\lbrace 0\rbrace $ and two distinct vertices $ I $ and $ J $ are adjacent if and only if $ IJ=0 $. In this paper, we characterize all positive integers $n$ for which $AG(\mathbb{Z}_n)$ is identifiable.Some properties of the generalized sierpi\'{n}ski gasket graphs
https://toc.ui.ac.ir/article_28245.html
The generalized Sierpiński gasket graphs $S[G,t]$ are introduced as the graphs obtained from the Sierpiński graphs $S(G,t)$ by contracting single edges between copies of previous phases. The family $S[G,t]$ is a generalization of a previously studied class of generalized Sierpiński gasket graphs $S[n,t]$. In this paper, several properties of $S[G,t]$ are studied. In particular, adjacency of vertices, degree sequence, general first Zagreb index, hamiltonicity, and Eulerian.On metric dimension of edge comb product of vertex-transitive graphs
https://toc.ui.ac.ir/article_28255.html
Suppose finite graph $G$ is simple, undirected and connected. If $W$ is an ordered set of the vertices such that $|W| = k$, the representation of a vertex $v$ is an ordered $k$-tuple consisting distances of vertex $v$ with every vertices in $W$. The set $W$ is defined as resolving vertex of $G$ if the $k$-tuples of every two vertices are distinct. Metric dimension of $G$, which is denoted by $dim(G)$, is the lowest size of $W$. In this paper, we provide a sharp lower bound of metric dimension for edge comb product graphs $G \cong T$ ▷e $H$ where $T$ is a tree graph and $H$ is a vertex-transitive graph. Moreover, we determine the exact value of metric dimension for edge comb product graphs $G \cong T$ ▷e $Ci_n(1,2)$ where $Ci_n(1,2)$ is a circulant graph.Density-Based clustering in mapReduce with guarantees on parallel time, space, and solution quality
https://toc.ui.ac.ir/article_28264.html
A well-known clustering problem called Density-Based Spatial Clustering of Applications with Noise~(DBSCAN) involves computing the solutions of at least one disk range query per input point, computing the connected components of a graph, and bichromatic fixed-radius nearest neighbor. MapReduce class is a model where a sublinear number of machines, each with sublinear memory, run for a polylogarithmic number of parallel rounds.&nbsp;Most of these problems either require quadratic time in the sequential model or are hard to compute in a constant number of rounds in MapReduce. In the Euclidean plane, DBSCAN algorithms with near-linear time and a randomized parallel algorithm with a polylogarithmic number of rounds exist.&nbsp;We solve DBSCAN in the Euclidean plane in a constant number of rounds in MapReduce, assuming the minimum number of points in range queries is constant and each connected component fits inside the memory of a single machine and has a constant diameter.Minimal graphs with respect to the multiplicative version of some vertex-degree-based topological indices
https://toc.ui.ac.ir/article_28369.html
As a real-valued function, a graphical parameter is defined on the class of finite simple graphs, and remains invariant under graph isomorphism. In mathematical chemistry, vertex-degree-based topological indices are the graph parameters of the general form of $p_{\phi}(G)=\sum_{uv\in E(G)}\phi(d(u),d(v))$, where $\phi$ represents a real-valued symmetric function, and $d(u)$ shows the degree of $u\in V(G)$. In this paper, it is proved that if $\phi$ has certain conditions, then the graph among those with $n$ vertices and $m$ edges, whose difference between the maximum and minimum degrees is at most $1$, has the minimal value of $p_{\phi}$. Moreover, it is demonstrated that some well-known topological indices are able to satisfy these certain conditions, and the given indices can be treated in a unified manner.On the inverse mostar index problem for molecular graphs
https://toc.ui.ac.ir/article_28370.html
Mostar indices are recently proposed distance-based graph invariants, that already have been much investigated and found applications. In this paper, we investigate the inverse problem for Mostar indices of unicyclic and bicyclic molecular graphs. We prove that all positive integers other than 1, 2, 3, and 5 can be the Mostar index of some bicyclic molecular graph. In addition, we resolve the inverse edge Mostar index problem for molecular unicyclic and bicyclic graphs, and in doing so, establish the second and third smallest value of the edge Mostar index of unicyclic graphs.A remark on sequentially Cohen-Macaulay monomial ideals
https://toc.ui.ac.ir/article_28426.html
Let $R=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$. We show that if $G$ is a connected graph with a basic $5$-cycle $C$, then $G$ is a sequentially Cohen-Macaulay graph if and only if there exists a shedding vertex $x$ of $C$ such that $G\setminus x$ and $G\setminus N[x]$ are sequentially Cohen-Macaulay graphs. Furthermore, we study the sequentially Cohen-Macaulay and Castelnuovo-Mumford regularity of square-free monomial ideals in some special cases.Forbidden subgraphs of Co-prime graphs of finite groups
https://toc.ui.ac.ir/article_28505.html
For a finite group $G$ the co-prime graph $\Gamma(G)$ is defined as a graph with vertex set $G$ in which two distinct vertices $x$ and $y$ are adjacent if and only if $gcd(o(x),o(y))=1$ where $o(x)$ and $o(y)$ denote the orders of the elements $x$ and $y$ respectively. In this paper we find properties of groups whose co-prime graphs forbid graphs such as $C_4,K_{1,3},P_4$ and asteroidal triples.On the indices of certain graph products
https://toc.ui.ac.ir/article_28578.html
Molecular descriptors are numerical graph invariants that are used to study the chemical structure of molecules. In this paper, we determine the upper bound of the Sombor index based on four operations involving the subdivision graph, semi-total point graph, semi-total line graph, and total graph related to the lexicographic and tensor product. The exact expressions of the first reformulated Zagreb index and the second hyper-Zagreb index of the tensor product are formulated on the basis of the four significant graphs. Further, the descriptors for certain standard graphs are obtained and the graphical comparison for the first reformulated Zagreb index has also been illustrated to understand the result better.Induced Geodetic Sequence of a Graph
https://toc.ui.ac.ir/article_28592.html
A vertex subset $S$ of a graph $G=(V,E)$ is said to be a geodetic set if every vertex in $G$ is in some $u-v$ geodesic for any $u,v \in S$. The minimum cardinality of such a set is the geodetic number, which is denoted as $g(G)$. In this paper, we introduce the concepts of induced geodetic number and induced geodetic sequence of a graph. We discuss this concept in some graph classes. Also, established the characterization of induced geodetic sequences for trees, unicyclic graphs and cacti.Star-Critical connected ramsey numbers for 2-colorings of complete graphs
https://toc.ui.ac.ir/article_28651.html
This paper builds upon Sumner's work by further investigating the concept of connected Ramsey numbers, specifically focusing on star-critical connected Ramsey numbers. We obtain star-critical connected Ramsey numbers for several cases of trees versus complete graphs, stars versus stars, and paths versus paths. The connected Ramsey number for a star versus $K_3$ is also evaluated. Exact values are also obtained for the connected Ramsey numbers of $K_{1,n}$ versus $K_3$. This research explores the interplay between connectivity and graph coloring within the context of Ramsey theory.Some results on $\lambda$-design conjecture
https://toc.ui.ac.ir/article_28652.html
Let $v$ and $\lambda$ be integers with $0&lt;\lambda&lt;v$. A $\lambda$-design $D$ is a pair $(X, \mathcal{A})$, where $X$ is a finite set with $v$ elements called points and $\mathcal{A}$ is a family of subsets of $X$ called blocks, with $|\mathcal{A}|=|X|$ such that&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (1) for all $B_i, B_j\in \mathcal{A},$ $i\neq j,$ $|B_i\cap B_j|=\lambda$;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (2) for all $B_j\in \mathcal{A},$ $|B_j|=k_j&gt;\lambda$, and not all $k_j$ are equal.The only known examples of $\lambda$-designs are so called of type-1 designs, which are obtained from symmetric designs by a certain complementation procedure. Ryser and Woodall had independently conjectured that all $\lambda$-designs are of type-1. Suppose $r$ and $r^*(r&gt;r^*)$ are replication numbers of $D$ and for distinct points $x$ and $y$ of $D$, let $\lambda(x,y)$ denote the number of blocks of $X$ containing $x$ and $y$.&nbsp;In this paper we investigate the possibilities of $\lambda$-designs to be of type-1 under the condition that $|\lambda(x,y)-\lambda(x,y')|&lt; 2 \left(\dfrac{r-r^*}{r+r^*-2}\right)$. Under this condition, we prove that if $ \dfrac{r-1}{r^*-1} \le 3$, then $\lambda$-design $D$ is of type-1. Also we prove that $D$ has exactly two distinct block sizes.The orders of subgroup products and coset products
https://toc.ui.ac.ir/article_28670.html
A sect is a subset of a group given by the product of a finite number of subgroups. It is generally not a direct product nor even a subgroup of the group. For finite groups, the orders of sects are their basic invariants. In this paper we describe properties of the orders of sects, such as divisibility and inequalities, which give constraints on the possible values of the orders of sects. We further consider clans, which are subsets of groups given by products of finite numbers of cosets. We also describe properties of their orders.On relationship between reformulated Sombor and other vertex--degree indices
https://toc.ui.ac.ir/article_28728.html
Let $G=(V,E)$, $V=\{v_1,v_2,\ldots, v_n\}$, $E=\{e_1,e_2, \ldots , e_m\}$, be a simple connected graph with $n\ge 2$ vertices and $m$ edges, with vertex degree sequence $\Delta=d_1\ge d_2\ge \cdots \ge d_n=\delta$, $d_i=d(v_i)$, and edge degree sequence $\Delta_e=d(e_1)\ge d(e_2)\ge \cdots \ge d(e_n)=\delta_e$. The reformulated Sombor index is defined as $RS(G) =\sum_{e_i\sim e_j}\sqrt{d(e_i)^2+d(e_j)^2}$. We consider a relationship between reformulated Sombor index and some of the vertex--degree-based indices.