Transactions on CombinatoricsTransactions on Combinatorics
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Feed provided by Transactions on Combinatorics. Click to visit.Sufficient conditions for triangle-free graphs to be super-$λ'$
http://toc.ui.ac.ir/article_22415_0.html
An edge-cut $F$ of a connected graph $G$ is called a‎ ‎restricted edge-cut if $G-F$ contains no isolated vertices‎. ‎The minimum cardinality of all restricted edge-cuts‎ ‎is called the restricted edge-connectivity $λ'(G)$ 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 super-$λ'$ if every minimum restricted edge-cut isolates‎ ‎an edge‎. ‎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, ‎Discrete Math‎., ‎310 (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-$λ'$‎, Network}, 309 (2009) 3336--3345]‎. Thu, 04 Jan 2018 20:30:00 +0100$\mathcal{B}$-partitions, determinant and permanent of graphs
http://toc.ui.ac.ir/article_22426_0.html
Let $G$ be a graph (directed or undirected) having $k$ number of blocks $B_1, B_2,hdots,B_k$. A $mathcal{B}$-partition of $G$ is a partition consists of $k$ vertex-disjoint subgraph $(hat{B_1},hat{B_1},hdots,hat{B_k})$ such that $hat{B}_i$ is an induced subgraph of $B_i$ for $i=1,2,hdots,k.$ 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 $(hat{B_1},hat{B_1},hdots,hat{B_k})$. 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.Tue, 20 Feb 2018 20:30:00 +0100The annihilator graph of a 0-distributive lattice
http://toc.ui.ac.ir/article_22285_4082.html
‎‎In 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 $Z(mathcal L)neqlbrace 0rbrace$‎, ‎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.Fri, 31 Aug 2018 19:30:00 +0100A spectral excess theorem for digraphs with normal Laplacian matrices
http://toc.ui.ac.ir/article_22346_4082.html
The 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 $g=3$‎.Fri, 31 Aug 2018 19:30:00 +0100