Transactions on Combinatorics
https://toc.ui.ac.ir/
Transactions on Combinatoricsendaily1Wed, 01 Dec 2021 00:00:00 +0330Wed, 01 Dec 2021 00:00:00 +0330Forcing edge detour monophonic number of a graph
https://toc.ui.ac.ir/article_25622.html
&lrm;For a connected graph $G=(V,E)$ of order at least two&lrm;, &lrm;an edge detour monophonic set&nbsp;of $G$ is a set $S$ of vertices such that every edge of $G$ lies on a detour monophonic path joining some pair of vertices in $S$&lrm;. &lrm;The edge detour monophonic number&nbsp;of $G$ is the minimum cardinality of its edge detour monophonic sets and is denoted by $edm(G)$&lrm;. &lrm;A subset $T$ of $S$ is a forcing edge detour monophonic subset&nbsp;for $S$ if $S$ is the unique edge detour monophonic set of size $edm(G)$ containing $T$&lrm;. &lrm;A forcing edge detour monophonic subset for $S$ of minimum cardinality is a minimum forcing edge detour monophonic subset&nbsp;of $S$&lrm;. &lrm;The forcing edge detour monophonic number&nbsp;$f_{edm}(S)$ in $G$ is the cardinality of a minimum forcing edge detour monophonic subset of $S$&lrm;. &lrm;The forcing edge detour monophonic number&nbsp;of $G$ is $f_{edm}(G)=min\{f_{edm}(S)\}$&lrm;, &lrm;where the minimum is taken over all edge detour monophonic sets $S$ of size $edm(G)$ in $G$&lrm;. &lrm;We determine bounds for it and find the forcing edge detour monophonic number of certain classes of graphs&lrm;. &lrm;It is shown that for every pair a&lrm;, &lrm;b&nbsp;of positive integers with $0\leq a&lt;b$ and $b\geq 2$&lrm;, &lrm;there exists a connected graph $G$ such that $f_{edm}(G)=a$ and $edm(G)=b$&lrm;.The Varchenko determinant of an oriented matroid
https://toc.ui.ac.ir/article_25621.html
Varchenko introduced in 1993 a distance function on the chambers of a hyperplane arrangement that gave rise to a determinant whose entry in position $(C, D)$ is the distance between the chambers $C$ and $D$, and computed that determinant. In 2017, Aguiar and Mahajan provided a generalization of that distance function, and computed the corresponding determinant. This article extends their distance function to the topes of an oriented matroid, and computes the determinant thus defined. Oriented matroids have the nice property to be abstractions of some mathematical structures including hyperplane and sphere arrangements, polytopes, directed graphs, and even chirality in molecular chemistry. Independently and with another method, Hochst\"{a}ttler and Welker also computed in 2019 the same determinant.Convolution identities involving the central binomial coefficients and Catalan numbers
https://toc.ui.ac.ir/article_25607.html
We generalize some convolution identities due to Witula and Qi et al&lrm;. &lrm;involving the central binomial coefficients and Catalan numbers&lrm;. &lrm;Our formula allows us to establish many new identities involving these important quantities&lrm;, &lrm;and recovers some known identities in the literature&lrm;. &lrm;Also&lrm;, &lrm;we give new proofs of Shapiro's Catalan convolution and a famous identity of Haj\'{o}s&lrm;.On the extremal connective eccentricity index among trees with maximum degree
https://toc.ui.ac.ir/article_25656.html
The connective eccentricity index (CEI) of a graph $G$ is defined as $\xi^{ce}(G)=\sum_{v \in V(G)}\frac{d_G(v)}{\varepsilon_G(v)}$, where $d_G(v)$ is the degree of $v$ and $\varepsilon_G(v)$ is the eccentricity of $v$. In this paper, we characterize the unique trees with the maximum and minimum CEI among all $n$-vertex trees and $n$-vertex conjugated trees with fixed maximum degree, respectively.On finite groups all of whose bi-Cayley graphs of bounded valency are integral
https://toc.ui.ac.ir/article_25652.html
Let $k\geq 1$ be an integer and $\mathcal{I}_k$ be&lrm; &lrm;the set of all finite groups $G$ such that every bi-Cayley graph BCay(G,S) of $G$ with respect to&lrm;&nbsp;&lrm;subset $S$ of length $1\leq |S|\leq k$ is integral&lrm;. &lrm;Let $k\geq 3$&lrm;. &lrm;We prove that a finite group $G$ belongs to $\mathcal{I}_k$ if and&lrm; &lrm;only if $G\cong\Bbb Z_3$&lrm;, &lrm;$\Bbb Z_2^r$ for some integer $r$&lrm;, &lrm;or $S_3$&lrm;.An effective new heuristic algorithm for solving permutation flow shop scheduling problem
https://toc.ui.ac.ir/article_25419.html
The deterministic permutation flow shop scheduling problem with makespan criterion is not solvable in polynomial time&lrm;. &lrm;Therefore&lrm;, &lrm;researchers have thought about heuristic algorithms&lrm;. &lrm;There are many heuristic algorithms for solving it that is a very important combinatorial optimization problem&lrm;. &lrm;In this paper&lrm;, &lrm;a new algorithm is proposed for solving the mentioned problem&lrm;. &lrm;The presented algorithm chooses the weighted path that starts from the up-left corner and reaches the down-right in the matrix of jobs processing times and calculates the biggest sum of the times in the footprints of this path&lrm;. &lrm;The row with the biggest sum permutes among all the rows of the matrix for locating the minimum of makespan&lrm;. &lrm;This method was run on Taillard&rsquo;s standard benchmark and the solutions were compared with the optimum or the best ones as well as 14 famous heuristics&lrm;. &lrm;The validity and effectiveness of the algorithm are shown with tables and statistical evaluation&lrm;.Vertex decomposability of complexes associated to forests
https://toc.ui.ac.ir/article_25654.html
In this article&lrm;, &lrm;we discuss the vertex decomposability of three well-studied simplicial complexes associated to forests&lrm;. &lrm;In particular&lrm;, &lrm;we show that the bounded degree complex of a forest and the complex of directed trees of a multidiforest is vertex decomposable&lrm;. &lrm;We then prove that the non-cover complex of a forest is either contractible or homotopy equivalent to a sphere&lrm;. &lrm;Finally we provide a complete characterization of forests whose non-cover complexes are vertex decomposable&lrm;.On the VC-dimension, covering and separating properties of the cycle and spanning tree hypergraphs of graphs
https://toc.ui.ac.ir/article_25746.html
In this paper&lrm;, &lrm;we delve into studying some relations between the structure of the cycles and spanning trees of a graph through the lens of its cycle and spanning tree hypergraphs which are hypergraphs with the edge set of the graph as their vertices and the edge sets of the cycles and spanning trees as their hyperedges respectively&lrm;. &lrm;In particular&lrm;, &lrm;we investigate relations between these hypergraphs from the perspective of the VC-dimension and some important separating and covering features of hypergraph theory and amongst the results&lrm;, &lrm;for example show that the VC-dimension of the cycle hypergraph is less than or equal to the VC-dimension of the spanning tree hypergraph and their gap can be arbitrary large. Note that VC-dimension is an important measure of complexity and a fundamental notion in numerous fields such as extremal combinatorics&lrm;, &lrm;graph theory&lrm;, &lrm;statistics and the theory of machine learning&lrm;. &lrm;Also we compare the separating and covering features of the mentioned hypergraphs and for instance show that the separating number of the cycle hypergraph is less than or equal to that of the spanning tree hypergraph&lrm;. &lrm;These hypergraphs help us to make several connections between cycles and spanning trees of graphs and compare their complexities&lrm;.