Transactions on CombinatoricsTransactions on Combinatorics
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Feed provided by Transactions on Combinatorics. Click to visit.$H$-kernels by walks in subdivision digraph
http://toc.ui.ac.ir/article_24341_4442.html
Let $H$ be a digraph possibly with loops and $D$ a digraph without loops whose arcs are colored with the vertices of $H$ ($D$ is said to be an $H$-colored digraph)‎. ‎A directed walk $W$ in $D$ is said to be an $H$-walk if and only if the consecutive colors encountered on $W$ form a directed walk in $H$‎. ‎A subset $N$ of the vertices of $D$ is said to be an $H$-kernel by walks if (1) for every pair of different vertices in $N$ there is no $H$-walk between them ($N$ is $H$-independent by walks) and (2) for each vertex $u$ in $V$($D$)-$N$ there exists an $H$-walk from $u$ to $N$ in $D$ ($N$ is $H$-absorbent by walks)‎. ‎Suppose that $D$ is a digraph possibly infinite‎. ‎In this paper we will work with the subdivision digraph $S_H$($D$) of $D$‎, ‎where $S_H$($D$) is an $H$-colored digraph defined as follows‎: ‎$V$($S_H$($D$)) = $V$($D$) $cup$ $A$($D$) and $A$($S_H$($D$)) = {($u$,$a$)‎ : ‎$a$ = ($u$,$v$) $in$ $A$($D$)} $cup$ {($a$,$v$)‎ : ‎$a$ = ($u$,$v$) $in$ $A$($D$)}‎, ‎where ($u$‎, ‎$a$‎, ‎$v$) is an $H$-walk in $S_H$($D$) for every $a$ = ($u$,$v$) in $A$($D$)‎. ‎We will show sufficient conditions on $D$ and on $S_H$($D$) which guarantee the existence or uniqueness of $H$-kernels by walks in $S_H$($D$)‎.Sat, 04 Jan 2020 20:30:00 +0100Connected zero forcing sets and connected propagation time of graphs
http://toc.ui.ac.ir/article_24400_0.html
The zero forcing number $Z(G)$ of a graph $G$is the minimum cardinality of a set $S$ with colored (black) vertices which forces the set $V(G)$to be colored (black) after some times. ``color change rule'': a white vertex is changed to a blackvertex when it is the only white neighbor of a black vertex.In this case, we say that the black vertex forces the white vertex.In this paper, we investigate the concept of connected zero forcing setand connected zero forcing number. We discusses this subject for special graphs and some products of graphs.Also we introduce the connected propagation time. Graphs with extreme minimum connected propagationtimes and maximum propagation times $|G|-1$ and $|G|-2$ are characterized.Sat, 18 Jan 2020 20:30:00 +0100Determinant Identities for Toeplitz-Hessenberg Matrices with Tribonacci Entries
http://toc.ui.ac.ir/article_24437_0.html
In this paper, we evaluate determinants of some families of Toeplitz--Hessenberg matrices having tribonacci number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers. In particular, we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz--Hessenberg determinants. We then obtain, by combinatorial arguments, extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying a recurrence of the form T_n^{(r)}=T_{n-1}^{(r)}+T_{n-2}^{(r)}+T_{n-r}^{(r)} for n >= r, with the appropriate initial conditions, where r >= 3 is arbitrary.Sun, 02 Feb 2020 20:30:00 +0100ON QUADRILATERALS IN THE SUBORBITAL GRAPHS OF THE NORMALIZER
http://toc.ui.ac.ir/article_24454_0.html
In this paper, we investigate suborbital graphs formed by N(Gamma_0(N)-invariant equivalencerelation induced on Q^. Conditions for being an edge are obtained as a main tool, then necessaryand sufficient conditions for the suborbital graphs to contain a circuit are investigated.Fri, 07 Feb 2020 20:30:00 +0100The distance spectrum of two new operations of graphs
http://toc.ui.ac.ir/article_24467_0.html
Let $G$ be a connected graph with vertex set $V(G)={v_1,v_2,cdots,v_n}$. The distance matrix $D=D(G)$ of $G$ is defined so that its $(i,j)$-entry is equal to the distance $d_G(v_i,v_j)$ between the vertices $v_i$ and $v_j$ of $G$. The eigenvalues ${mu_1,mu_2,cdots,mu_n}$ of $D(G)$ are the $D$-eigenvalues of $G$ and form the distance spectrum or the $D$-spectrum of $G$, denoted by $Spec_D(G)$. In this paper, we introduce two new operations $G_1blacksquare_k G_2$ and $G_1blacklozenge_k G_2$ on graphs $G_1$ and $G_2$, and describe the distance spectra of $G_1blacksquare_k G_2$ and $G_1blacklozenge_k G_2$ of regular graphs $G_1$ and $G_2 $ in terms of their adjacency spectra. By using these results, we obtain some new integral adjacency spectrum graphs, integral distance spectrum graphs and a number of families of sets of noncospectral graphs with equal distance energy.Wed, 12 Feb 2020 20:30:00 +0100Zero-Sum Flow Number of Categorical and Strong Product of Graphs
http://toc.ui.ac.ir/article_24517_0.html
A zero-sum flow is an assignment ofnonzero integers to the edges such that the sum of the values of alledges incident with each vertex is zero, and we call it a zero-sumk-flow if the absolute values of edges are less than k. Wedefine the zero-sum flow number of G as the least integer k forwhich G admitting a zero sum k-flow.In this paper we gave complete zero-sum flow and zero sum numbersfor categorical and strong product of two graphs namely cycle and paths.Thu, 05 Mar 2020 20:30:00 +0100The diameter of the non-nilpotent graph of a finite group
http://toc.ui.ac.ir/article_24553_0.html
We prove that the graph obtained from the non-nilpotent graph of a finite group by deleting the isolated vertices is connected with diameter at most 3. This bound is the best possible.Fri, 17 Apr 2020 19:30:00 +0100Further results on maximal rainbow domination number
http://toc.ui.ac.ir/article_24579_0.html
A {em 2-rainbow dominating function} (2RDF) of a graph $G$ is afunction $f$ from the vertex set $V(G)$ to the set of all subsetsof the set ${1,2}$ such that for any vertex $vin V(G)$ with$f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$is fulfilled, where $N(v)$ is the open neighborhood of $v$. A {emmaximal 2-rainbow dominating function} of a graph $G$ is a2-rainbow dominating function $f$ such that the set ${winV(G)|f(w)=emptyset}$ is not a dominating set of $G$. The {emweight} of a maximal 2RDF $f$ is the value $omega(f)=sum_{vinV}|f (v)|$. The {em maximal $2$-rainbow domination number} of agraph $G$, denoted by $gamma_{m2r}(G)$, is the minimum weight of amaximal 2RDF of $G$. In this paper, we continue the study of maximal2-rainbow domination {number} in graphs. Specially, we first characterize all graphs with largemaximal 2-rainbow domination number. Finally, we determine the maximal 2-rainbow domination number in the sun and sunlet graphs.Wed, 29 Apr 2020 19:30:00 +0100The Vertex Steiner Number of a Graph
http://toc.ui.ac.ir/article_24580_0.html
Let x be a vertex of a connected graph G and W ⊂V(G) such that x∉W.Then W is called an x - Steiner set of G if W⋃{x} is a steiner set of G. The minimum cardinality of an x - Steiner set of G is defined as x - Steiner number of G and denoted by s_x (G). Some general properties satisfied by this concept are studied. The x - Steiner numbers of certain classes of graphs are determined. Connected graphs of order p with x - Steiner number 1 or p-1 are characterized. It is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G such that s(G) = a and s_x (G)= b for some vertex x in G, where s(G) is the Steiner number of a graph.Wed, 29 Apr 2020 19:30:00 +0100Exact bounds for (λ, n)–stable 0-1 matrices.
http://toc.ui.ac.ir/article_24585_0.html
Consider a v × v (0, 1) matrix A with exactly k ones in eachrow and each column. A is (λ, n)–stable, if it does not contain anyλ × n submatrix with exactly one 0. If A is (λ, n)–stable, λ, n ≥ 2,then under suitable conditions on A, v ≥ k k(n−1)+(λ−2) . The case n λ−2of equality leads to new and substantive connections with block designs. The previous bound and characterization of (λ, 2)–stable matrices follows immediately as a special case.Sat, 02 May 2020 19:30:00 +0100