University of IsfahanTransactions on Combinatorics2251-865710420211201Forcing edge detour monophonic number of a graph2012112562210.22108/toc.2021.119182.1670ENP.TitusDepartment of Mathematics, University College of Engineering Nagercoil, Nagercoil-629 004, IndiaK.GanesamoorthyDepartment of Mathematics, Coimbatore Institute of Technology, Coimbatore - 641 014, IndiaJournal Article20190915For 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 $0\leq a<b$ and $b\geq 2$, there exists a connected graph $G$ such that $f_{edm}(G)=a$ and $edm(G)=b$.https://toc.ui.ac.ir/article_25622_c063c926a1ab86fa5bf537a4b45903ac.pdfUniversity of IsfahanTransactions on Combinatorics2251-865710420211201The Varchenko determinant of an oriented matroid2132242562110.22108/toc.2021.125990.1780ENHeryRandriamaroLot II B 32 bis Faravohitra, 101 Antananarivo, MadagascarJournal Article20201113Varchenko 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.pdfUniversity of IsfahanTransactions on Combinatorics2251-865710420211201Convolution identities involving the central binomial coefficients and Catalan numbers2252382560710.22108/toc.2021.127505.1821ENNecdetBatırDepartment of Mathematics, Nev¸ sehir Hacı Bekta¸ s Veli University, 50300, Nev¸ sehir, TurkeyHakanKucukDepartment of Mathematics, Nev¸ sehir Hacı Bekta¸ s Veli University, 50300, Nev¸ sehir, TurkeySezerSorgunDepartment of Mathematics, Nev¸ sehir Hacı Bekta¸ s Veli University, 50300, Nev¸ sehir, Turkey0000-0001-8708-1226Journal Article20210220We 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_dd43ffb28902288a41a5d2ea54377359.pdfUniversity of IsfahanTransactions on Combinatorics2251-865710420211201On the extremal connective eccentricity index among trees with maximum degree2392462565610.22108/toc.2021.120679.1693ENFazalHayatSchool of Mathematical Sciences, South China Normal University,
Guangzhou 510631, PR ChinaJournal Article20191223The 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.pdfUniversity of IsfahanTransactions on Combinatorics2251-865710420211201On finite groups all of whose bi-Cayley graphs of bounded valency are integral2472522565210.22108/toc.2021.126275.1787ENMajidArezoomandUniversity of Larestan, 74317-16137, Lar, IranJournal Article20201201Let $k\geq 1$ 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 $1\leq |S|\leq k$ is integral. Let $k\geq 3$. We prove that a finite group $G$ belongs to $\mathcal{I}_k$ if and only if $G\cong\Bbb Z_3$, $\Bbb Z_2^r$ for some integer $r$, or $S_3$.https://toc.ui.ac.ir/article_25652_0216d6b0de713ce4fb52bdfe4e1afc65.pdf