University of IsfahanTransactions on Combinatorics2251-865715120241226Equable kites, trapezoids and cyclic quadrilaterals on the Eisenstein Lattice1112909510.22108/toc.2024.140684.2150ENChristianAebiCollège Calvin Geneva, Switzerland0000-0002-8667-5214GrantCairnsDepartment of Mathematical and Physical Sciences, La Trobe University Melbourne, AustraliaJournal Article20240211We show that on the Eisenstein lattice, up to Euclidean motions, there is only one infinite family of equable kites, which is given by the Pell-like equation $3x^2-2=y^2$, and only one single equable trapezoid, which also happens to be the only equable cyclic quadrilateral.https://toc.ui.ac.ir/article_29095_00f0c55302d8d4bcfeb3d86338686f3a.pdfUniversity of IsfahanTransactions on Combinatorics2251-865715120241225The first and second Zagreb indices of hypergraphs13282909610.22108/toc.2024.141216.2169ENWeiGaoDepartment of Mathematics, Pennsylvania State University at Abington, Abington, PA, 19001, USAJournal Article20240418Let $\mathcal{H}$ be a hypergraph on the non-empty finite vertex set $V(\mathcal{H})$ with the hyperedge set $E(\mathcal{H})$, where each hyperedge $e\in E(\mathcal{H})$ is a subset of $V(\mathcal{H})$ with at least two vertices.<br />The bounds on the first and second Zagreb indices of hypergraphs, weak bipartite hypergraphs, hypertrees, $k$-uniform hypergraphs, $k$-uniform weak bipartite hypergraphs, and $k$-uniform hypertrees are discussed.https://toc.ui.ac.ir/article_29096_4dc09c0693b5d4275fbce8f051069f98.pdfUniversity of IsfahanTransactions on Combinatorics2251-865715120250106The non-two-primes graph of a finite group29362913910.22108/toc.2025.142373.2201ENKarmeleGaratea-ZaballaDipartimento di Matematica “Tullio Levi Civita”, Università di Padova, Via Trieste 63, 35121 Padova, ItalyAndreaLucchiniDipartimento di Matematica “Tullio Levi Civita”, Università di Padova, Via Trieste 63, 35121 Padova, ItalyJournal Article20240807To any finite group $G$, we may associate a graph whose vertices are the elements of $G$ and where two distinct vertices $x$ and $y$ are adjacent if and only if the order of the subgroup $\langle x, y\rangle$ is divisible by at least 3 distinct primes. We prove that the subgraph of this graph induced by the non-isolated vertices is connected and has diameter at most 5.https://toc.ui.ac.ir/article_29139_237b6dbf9241b56f1cdd9a37d331898c.pdfUniversity of IsfahanTransactions on Combinatorics2251-865715120250110Automorphism group of a family of distance-regular graphs which are not distance-transitive37462914010.22108/toc.2025.142386.2200ENSeyed MortezaMirafzalDepartment of Mathematics, Lorestan University, Khorramabad, IranAngsumanDasDepartment of Mathematics, Presidency University, Kolkata, IndiaJournal Article20240806Let $G_n=\mathbb{Z}_n\times \mathbb{Z}_n$ for $n\geq 4$ and $S=\{(i,0),(0,i),(i,i): 1\leq i \leq n-1\}\subset G_n$. Define $\Gamma(n)$ to be the Cayley graph of $G_n$ with respect to the connecting set $S$. It is known that $\Gamma(n)$ is a strongly regular graph with the parameters $(n^2, 3n-3, n, 6)$ \cite{19}. Hence $\Gamma(n)$ is a distance-regular graph. It is known that every distance-transitive graph is distance-regular, but the converse is not true. In this paper, we study some algebraic properties of the graph $\Gamma(n)$. Then by determining the automorphism group of this family of graphs, we show that the graphs under study are not distance-transitive.https://toc.ui.ac.ir/article_29140_dd6695eed070fee5dbeecc9be6d555e2.pdfUniversity of IsfahanTransactions on Combinatorics2251-865715120250218Some designs from the fixed points of alternating groups47542926110.22108/toc.2025.141435.2178ENMadimetja JanKekanaDepartment of Mathematics and Applied Mathematics, Faculty of Science and Agriculture, University of Limpopo, South Africa.AminSaeidiDepartment of mathematics and Applied Mathematics, University of Limpopo, South Africa.ThekisoSeretloDepartment of Pure and Applied Analytics, North West University, mafikeng Campus, Mmabatho, South Africa.Journal Article20240527In this paper, we construct some $1-(v,k,\lambda)$ designs from the alternating group $G=A_{n}$ with the maximal subgroup isomorphic to $M=A_{n-1}$. The method we use is called Key-Moori Method $2$. Furthermore, from the set $I_x$ which is the intersection of all blocks containing the point $x\in G$, we construct corresponding reduced designs. Our aim is to give explicit formulae to compute the parameters of the designs based on the cyclic structures of the permutations in $G$.https://toc.ui.ac.ir/article_29261_6774507586420233b8c31ee896f3a595.pdfUniversity of IsfahanTransactions on Combinatorics2251-865715120260301A characterization of graphs with upper locating-domination number equal to $n-2$55682927810.22108/toc.2025.139873.2128ENMalikaMimouniLaboratory, Department of Mathematics (LATSI), Faculty of Sciences, University of Blida 1, P.O.Box 270, Blida, AlgeriaLyesOuldrabahLaboratory of Mathematics and its Applications (LMA), Faculty of Technology, Medea University, Medea, Algeria0000-0002-8667-5214NoureddineIkhlef-EschoufLaboratory of Mathematics and its Applications (LMA), Faculty of Sciences, Medea University, Medea, AlgeriaJournal Article20231121A set $D$ of vertices in a graph $G$ is called a dominating set of $G$ if every vertex in $V\left( G\right) \backslash D$ has at least one neighbor in $D$. A dominating set $D$ of $G$ is called a locating-dominating set of $G$ if every two vertices in $V\left( G\right) \backslash D$ have two distinct neighborhood sets. The upper locating-domination number $\Gamma_{L}(G)$ is the maximum cardinality of a minimal locating-dominating set of $G.$ In this paper, we characterize the graphs with $\Gamma_{L}\left( G\right) =n-2$.https://toc.ui.ac.ir/article_29278_89a258cdc8a7903191ed89da8a6d0245.pdf