University of Isfahan Transactions on Combinatorics 2251-8657 10 4 2021 12 01 Forcing edge detour monophonic number of a graph 201 211 25622 10.22108/toc.2021.119182.1670 EN P. Titus Department of Mathematics, University College of Engineering Nagercoil, Nagercoil-629 004, India K. Ganesamoorthy Department of Mathematics, Coimbatore Institute of Technology, Coimbatore - 641 014, India Journal Article 2019 09 15 ‎For a connected graph \$G=(V,E)\$ of order at least two‎, ‎an <em>edge detour monophonic set</em> 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\$‎. ‎The <em>edge detour monophonic number</em> of \$G\$ is the minimum cardinality of its edge detour monophonic sets and is denoted by \$edm(G)\$‎. ‎A subset \$T\$ of \$S\$ is a <em>forcing edge detour monophonic subset</em> for \$S\$ if \$S\$ is the unique edge detour monophonic set of size \$edm(G)\$ containing \$T\$‎. ‎A forcing edge detour monophonic subset for \$S\$ of minimum cardinality is a <em>minimum forcing edge detour monophonic subset</em> of \$S\$‎. ‎The <em>forcing edge detour monophonic number</em> \$f_{edm}(S)\$ in \$G\$ is the cardinality of a minimum forcing edge detour monophonic subset of \$S\$‎. ‎The <em>forcing edge detour monophonic number</em> of \$G\$ is \$f_{edm}(G)=min{f_{edm}(S)}\$‎, ‎where the minimum is taken over all edge detour monophonic sets \$S\$ of size \$edm(G)\$ in \$G\$‎. ‎We determine bounds for it and find the forcing edge detour monophonic number of certain classes of graphs‎. ‎It is shown that for every pair <em>a</em>‎, ‎<em>b</em> of positive integers with \$0leq a<b\$ and \$bgeq 2\$‎, ‎there exists a connected graph \$G\$ such that \$f_{edm}(G)=a\$ and \$edm(G)=b\$‎. https://toc.ui.ac.ir/article_25622_c063c926a1ab86fa5bf537a4b45903ac.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 10 4 2021 12 01 The Varchenko determinant of an oriented matroid 213 224 25621 10.22108/toc.2021.125990.1780 EN Hery Randriamaro Lot II B 32 bis Faravohitra, 101 Antananarivo, Madagascar Journal Article 2020 11 13 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. https://toc.ui.ac.ir/article_25621_1c540fade1329af3d4282f58ec3b77e9.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 10 4 2021 12 01 Convolution identities involving the central binomial coefficients and Catalan numbers 225 238 25607 10.22108/toc.2021.127505.1821 EN Necdet Batır Department of Mathematics, Nev¸ sehir Hacı Bekta¸ s Veli University, 50300, Nev¸ sehir, Turkey Hakan Kucuk Department of Mathematics, Nev¸ sehir Hacı Bekta¸ s Veli University, 50300, Nev¸ sehir, Turkey Sezer Sorgun Department of Mathematics, Nev¸ sehir Hacı Bekta¸ s Veli University, 50300, Nev¸ sehir, Turkey 0000-0001-8708-1226 Journal Article 2021 02 20 We generalize some convolution identities due to Witula and Qi et al‎. ‎involving the central binomial coefficients and Catalan numbers‎. ‎Our formula allows us to establish many new identities involving these important quantities‎, ‎and recovers some known identities in the literature‎. ‎Also‎, ‎we give new proofs of Shapiro's Catalan convolution and a famous identity of Haj'{o}s‎. https://toc.ui.ac.ir/article_25607_6a04c76f9857026c8aa01f717763c446.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 10 4 2021 12 01 On the extremal connective eccentricity index among trees with maximum degree 239 246 25656 10.22108/toc.2021.120679.1693 EN Fazal Hayat School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China Journal Article 2019 12 23 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. https://toc.ui.ac.ir/article_25656_ce36f15a7dee2e1ace7fdfcaf03f97ae.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 10 4 2021 12 01 On finite groups all of whose bi-Cayley graphs of bounded valency are integral 247 252 25652 10.22108/toc.2021.126275.1787 EN Majid Arezoomand University of Larestan, 74317-16137, Lar, Iran Journal Article 2020 12 01 Let \$kgeq 1\$ be an integer and \$mathcal{I}_k\$ be‎ ‎the set of all finite groups \$G\$ such that every bi-Cayley graph BCay(G,S) of \$G\$ with respect to‎ ‎subset \$S\$ of length \$1leq |S|leq k\$ is integral‎. ‎Let \$kgeq 3\$‎. ‎We prove that a finite group \$G\$ belongs to \$mathcal{I}_k\$ if and‎ ‎only if \$GcongBbb Z_3\$‎, ‎\$Bbb Z_2^r\$ for some integer \$r\$‎, ‎or \$S_3\$‎. https://toc.ui.ac.ir/article_25652_0216d6b0de713ce4fb52bdfe4e1afc65.pdf