University of IsfahanTransactions on Combinatorics2251-86577320180901The annihilator graph of a 0-distributive lattice1182228510.22108/toc.2017.104919.1507ENSaeidBagheriDepartment of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, IranMahtabKoohi KerahroodiDepartment of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran.Journal Article20170629In this article, for a lattice $\mathcal L$, we define and investigate the annihilator graph $\mathfrak {ag} (\mathcal L)$ of $\mathcal L$ which contains the zero-divisor graph of $\mathcal L$ as a subgraph. Also, for a 0-distributive lattice $\mathcal L$, we study some properties of this graph such as regularity, connectedness, the diameter, the girth and its domination number. Moreover, for a distributive lattice $\mathcal L$ with $Z(\mathcal L)\neq\lbrace 0\rbrace$, we show that $\mathfrak {ag} (\mathcal L) = \Gamma(\mathcal L)$ if and only if $\mathcal L$ has exactly two minimal prime ideals. Among other things, we consider the annihilator graph $\mathfrak {ag} (\mathcal L)$ of the lattice $\mathcal L=(\mathcal D(n),|)$ containing all positive divisors of a non-prime natural number $n$ and we compute some invariants such as the domination number, the clique number and the chromatic number of this graph. Also, for this lattice we investigate some special cases in which $\mathfrak {ag} (\mathcal D(n))$ or $\Gamma(\mathcal D(n))$ are planar, Eulerian or Hamiltonian.https://toc.ui.ac.ir/article_22285_719ab505eba5ec2cd4bf741957e5ce29.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577320180901A spectral excess theorem for digraphs with normal Laplacian matrices19282234610.22108/toc.2018.105873.1513ENFatemeShafieiIsfahan University of TechnologyJournal Article20170811The spectral excess theorem, due to Fiol and Garriga in 1997, is an important result, because it gives a good characterization of distance-regularity in graphs. Up to now, some authors have given some variations of this theorem. Motivated by this, we give the corresponding result by using the Laplacian spectrum for digraphs. We also illustrate this Laplacian spectral excess theorem for digraphs with few Laplacian eigenvalues and we show that any strongly connected and regular digraph that has normal Laplacian matrix with three distinct eigenvalues, is distance-regular. Hence such a digraph is strongly regular with girth $g=2$ or $g=3$.https://toc.ui.ac.ir/article_22346_f0401337d3cc116dc87ace2c1fba2dc5.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577320180901Sufficient conditions for triangle-free graphs to be super-$λ'$29362241510.22108/toc.2018.106623.1523ENHuiwenChengDepartment of Mathematics, Zhongguancun InstituteYan-JingLiDepartment of Mathematics, Beijing Jiaotong University, ChinaJournal Article20170924An edge-cut $F$ of a connected graph $G$ is called a <em>restricted edge-cut</em> if $G-F$ contains no isolated vertices. The minimum cardinality of all restricted edge-cuts is called the <em>restricted edge-connectivity</em> $λ'(G)$ of $G$. A graph $G$ is said to be $λ'$-optimal if $λ'(G)=\xi(G)$, where $\xi(G)$ is the minimum edge-degree of $G$. A graph is said to be <em>super</em>-$λ'$ if every minimum restricted edge-cut isolates an edge.<br /> <br /> In this paper, first, we provide a short proof of a previous theorem about the sufficient condition for $λ'$-optimality in triangle-free graphs, which was given in [J. Yuan and A. Liu, Sufficient conditions for $λ_k$-optimality in triangle-free graphs, <em>Discrete Math.</em>, <strong>310</strong> (2010) 981--987]. Second, we generalize a known result about the sufficient condition for triangle-free graphs being super-$λ'$ which was given by Shang et al. in [L. Shang and H. P. Zhang, Sufficient conditions for graphs to be $λ'$-optimal and super-$λ'$, N<em>etwork</em>}, <strong>309</strong> (2009) 3336--3345].<br /> https://toc.ui.ac.ir/article_22415_297dcf89662c9125339a728e85ad39e3.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577320180901$\mathcal{B}$-Partitions, determinant and permanent of graphs37542242610.22108/toc.2017.105288.1508ENRanveerSinghDepartment of Mathematics, Indian Institute of Technology Jodhpur, Jodhpur, IndiaRavindra B.BapatStat-Math Unit, ISI DelhiJournal Article20170711Let $G$ be a graph (directed or undirected) having $k$ number of blocks $B_1, B_2,\ldots,B_k$. A $\mathcal{B}$-partition of $G$ is a partition consists of $k$ vertex-disjoint subgraph $(\hat{B_1},\hat{B_1},\ldots,\hat{B_k})$ such that $\hat{B}_i$ is an induced subgraph of $B_i$ for $i=1, 2,\ldots,k.$ The terms $\prod_{i=1}^{k}\det(\hat{B}_i),\ \prod_{i=1}^{k}\text{per}(\hat{B}_i)$ represent the det-summands and the per-summands, respectively, corresponding to the $\mathcal{B}$-partition $(\hat{B_1},\hat{B_1},\ldots,\hat{B_k})$. The determinant (permanent) of a graph having no loops on its cut-vertices is equal to the summation of the det-summands (per-summands), corresponding to all possible $\mathcal{B}$-partitions. In this paper, we calculate the determinant and the permanent of classes of graphs such as block graph, block graph with negatives cliques, signed unicyclic graph, mixed complete graph, negative mixed complete graph, and star mixed block graphs.https://toc.ui.ac.ir/article_22426_684cba1ff8383118f056e8041a6e743a.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577320180901Iota energy of weighted digraphs55732270710.22108/toc.2018.109248.1546ENSumairaHafeezSchool of Natural Sciences, National university of sciences and Technology Islamabad, PakistanMehtabKhanDepartment of mathematics, school of Natural Sciences, National University of Sciences and Technology Islamabad, PakistanJournal Article20180122The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. The iota energy of a digraph is recently defined as the sum of absolute values of imaginary part of its eigenvalues. In this paper, we extend the concept of iota energy of digraphs to weighted digraphs. We compute the iota energy formulae for the positive and negative weight directed cycles. We also characterize the unicyclic weighted digraphs with cycle weight $ r \in [-1, 1]\backslash \{0\}$ having minimum and maximum iota energy. We obtain well known McClelland upper bound for the iota energy of weighted digraphs. Finally, we find the class of noncospectral equienergetic weighted digraphs.https://toc.ui.ac.ir/article_22707_8eb3ec61aa18a4a5a2445c45716a4a23.pdf