University of IsfahanTransactions on Combinatorics2251-865712420231201Line graphs associated to annihilating-ideal graph attached to lattices of genus one1751902689710.22108/toc.2022.125344.1771ENAtossaParsapourDepartment of Mathematics, Bandar Abbas Branch, Islamic Azad University, Bandar Abbas, Iran0000000166077535KhadijehAhmadJavaheriDepartment of Mathematics, Bandar Abbas Branch, Islamic Azad University, Bandar Abbas, Iran0000000191857218Journal Article20201009Let $(L,\wedge,\vee)$ be a lattice with a least element $0$. The annihilating-ideal graph of $L$, denoted by $\mathbb{AG}(L)$, is a graph whose vertex-set is the set of all non-trivial ideals of $L$ and, for every two distinct vertices $I$ and $J$, the vertex $I$ is adjacent to $J$ if and only if $I\wedge J=\{0\}$. In this paper, we characterize all lattices $L$ whose the graph $\mathfrak{L}(\mathbb{AG}(L))$ is toroidal.https://toc.ui.ac.ir/article_26897_4663430929eaa6339d1f435f56d8c2d5.pdfUniversity of IsfahanTransactions on Combinatorics2251-865712420231201On the spectral radius, energy and Estrada index of the Sombor matrix of graphs1912052689610.22108/toc.2022.127710.1827ENZhenLinSchool of Mathematics and Statistics, Qinghai Normal University, 810008, Xining, P. R. China
The State Key Laboratory of Tibetan Intelligent Information Processing and Application, Qinghai Normal University,
810008, Xining, P. R. ChinaTingZhouSchool of Mathematics, China University of Mining and Technology, 221116, Xuzhou, P. R. ChinaLianyingMiaoSchool of Mathematics, China University of Mining and Technology, 221116, Xuzhou, P. R. ChinaJournal Article20210308Let $G$ be a simple undirected graph with vertex set $V(G)=\{v_1, v_2,\ldots,v_n\}$ and edge set $E(G)$. The Sombor matrix $\mathcal{S}(G)$ of a graph $G$ is defined so that its $(i,j)$-entry is equal to $\sqrt{d_i^2+d_j^2}$ if the vertices $v_i$ and $v_j$ are adjacent, and zero otherwise, where $d_i$ denotes the degree of vertex $v_i$ in $G$. In this paper, lower and upper bounds on the spectral radius, energy and Estrada index of the Sombor matrix of graphs are obtained, and the respective extremal graphs are characterized.https://toc.ui.ac.ir/article_26896_ab1f533d12c78afa9542037d171aed60.pdfUniversity of IsfahanTransactions on Combinatorics2251-865712420231201Distance (signless) Laplacian spectrum of dumbbell graphs2072162692310.22108/toc.2022.131435.1932ENSakthideviKaliyaperumalDivision of Nathenatics, School of Advanced Sciences, Vellore Institute of Technology, ChennaiKalyaniDesikanDivision of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai0000-0002-3074-5826Journal Article20211112In this paper, we determine the distance Laplacian and distance signless Laplacian spectrum of generalized wheel graphs and a new class of graphs called dumbbell graphs.https://toc.ui.ac.ir/article_26923_8e954821a49839faeba70d442968b295.pdfUniversity of IsfahanTransactions on Combinatorics2251-865712420231201On Laplacian resolvent energy of graphs2172252692210.22108/toc.2022.133236.1983ENSandeepBhatnagarDepartment of Applied Mathematics, Aligarh Muslim University, Aligarh, IndiaSiddiquiMerajuddinDepartment of Applied Mathematics, Aligarh Muslim University, Aligarh, IndiaShariefuddinPirzadaDepartment of Mathematics, University of Kashmir, Srinagar, IndiaJournal Article20220409Let $G$ be a simple connected graph of order $n$ and size $m$. The matrix $L(G)=D(G)-A(G)$ is the Laplacian matrix of $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix, respectively. For the graph $G$, let $d_{1}\geq d_{2}\geq \cdots d_{n}$ be the vertex degree sequence and $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{n-1}>\mu_{n}=0$ be the Laplacian eigenvalues. The Laplacian resolvent energy $RL(G)$ of a graph $G$ is defined as $RL(G)=\sum\limits_{i=1}^{n}\frac{1}{n+1-\mu_{i}}$. In this paper, we obtain an upper bound for the Laplacian resolvent energy $RL(G)$ in terms of the order, size and the algebraic connectivity of the graph. Further, we establish relations between the Laplacian resolvent energy $RL(G)$ with each of the Laplacian-energy-Like invariant $LEL$, the Kirchhoff index $Kf$ and the Laplacian energy $LE$ of the graph.https://toc.ui.ac.ir/article_26922_3f840846eddc3ba6b2d0c4a558b29495.pdfUniversity of IsfahanTransactions on Combinatorics2251-865712420231201General sum-connectivity index of trees with given number of branching vertices2272382700910.22108/toc.2022.133548.1987ENTomasVetrikDepartment of Mathematics and Applied Mathematics,
University of the Free State,
Bloemfontein, South AfricaJournal Article20220507In 2015, Borovi\'{c}anin presented trees with the smallest first Zagreb index among trees with given number of vertices and number of branching vertices. The first Zagreb index is obtained from the general sum-connectivity index if $a = 1$. For $a \in \mathbb{R}$, the general sum-connectivity index of a graph $G$ is defined as $\chi_{a} (G) = \sum_{uv\in E(G)} [d_G (u) + d_G (v)]^{a}$, where $E(G)$ is the edge set of $G$ and $d_G (v)$ is the degree of a vertex $v$ in $G$. We show that the result of Borovi\'{c}anin cannot be generalized for the general sum-connectivity index ($\chi_{a}$ index) if $0 < a < 1$ or $a > 1$. Moreover, the sets of trees having the smallest $\chi_a$ index are not the same for $0 < a < 1$ and $a > 1$. Among trees with given number of vertices and number of branching vertices, we present all the trees with the smallest $\chi_a$ index for $0 < a < 1$ and $a > 1$. Since the hyper-Zagreb index is obtained from the $\chi_a$ index if $a = 2$, results on the hyper-Zagreb index are corollaries of our results on the $\chi_a$ index for $a > 1$.https://toc.ui.ac.ir/article_27009_258b0c2b01967adee3222ab06f3b8e28.pdf