@Article{Dutta2017,
author="Dutta, Sanghita
and Lanong, Chanlemki",
title="On annihilator graph of a finite commutative ring",
journal="Transactions on Combinatorics",
year="2017",
volume="6",
number="1",
pages="1-11",
abstract="The annihilator graph $AG(R)$ of a commutative ring $R$ is a simple undirected graph with the vertex set $Z(R)^*$ and two distinct vertices are adjacent if and only if $ann(x) \cup ann(y)$ $ \neq $ $ann(xy)$. In this paper we give the sufficient condition for a graph $AG(R)$ to be complete. We characterize rings for which $AG(R)$ is a regular graph, we show that $\gamma (AG(R))\in \{1,2\}$ and we also characterize the rings for which $AG(R)$ has a cut vertex. Finally we find the clique number of a finite reduced ring and characterize the rings for which $AG(R)$ is a planar graph.",
issn="2251-8657",
doi="10.22108/toc.2017.20360",
url="http://toc.ui.ac.ir/article_20360.html"
}
@Article{Gao2017,
author="Gao, Yun
and Farahani, Mohammad Reza
and Gao, Wei",
title="A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs",
journal="Transactions on Combinatorics",
year="2017",
volume="6",
number="1",
pages="13-19",
abstract="A graph $G$ is called a fractional $(k,n',m)$-critical deleted graph if any $n'$ vertices are removed from $G$ the resulting graph is a fractional $(k,m)$-deleted graph. In this paper, we prove that for integers $k\ge 2$, $n',m\ge0$, $n\ge8k+n'+4m-7$, and $\delta(G)\ge k+n'+m$, if $$|N_{G}(x)\cup N_{G}(y)|\ge\frac{n+n'}{2}$$ for each pair of non-adjacent vertices $x$, $y$ of $G$, then $G$ is a fractional $(k,n',m)$-critical deleted graph. The bounds for neighborhood union condition, the order $n$ and the minimum degree $\delta(G)$ of $G$ are all sharp.",
issn="2251-8657",
doi="10.22108/toc.2017.20355",
url="http://toc.ui.ac.ir/article_20355.html"
}
@Article{Pirzada2017,
author="Pirzada, Shariefuddin
and A. Chat, Bilal",
title="The condition for a sequence to be potentially $A_{L, M}$- graphic",
journal="Transactions on Combinatorics",
year="2017",
volume="6",
number="1",
pages="21-27",
abstract="The set of all non-increasing non-negative integer sequences $\pi=(d_1, d_2,\ldots,d_n)$ is denoted by $NS_n$. A sequence $\pi\in NS_{n}$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices, and such a graph $G$ is called a realization of $\pi$. The set of all graphic sequences in $NS_{n}$ is denoted by $GS_{n}$. The complete product split graph on $L + M$ vertices is denoted by $\overline{S}_{L, M}=K_{L} \vee \overline{K}_{M}$, where $K_{L}$ and $K_{M}$ are complete graphs respectively on $L = \sum\limits_{i = 1}^{p}r_{i}$ and $M = \sum\limits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers. Another split graph is denoted by $S_{L, M} = \overline{S}_{r_{1}, s_{1}} \vee\overline{S}_{r_{2}, s_{2}} \vee \cdots \vee \overline{S}_{r_{p}, s_{p}}= (K_{r_{1}} \vee \overline{K}_{s_{1}})\vee (K_{r_{2}} \vee \overline{K}_{s_{2}})\vee \cdots \vee (K_{r_{p}} \vee \overline{K}_{s_{p}})$. A sequence $\pi=(d_{1}, d_{2},\ldots,d_{n})$ is said to be potentially $S_{L, M}$-graphic (respectively $\overline{S}_{L, M}$)-graphic if there is a realization $G$ of $\pi$ containing $S_{L, M}$ (respectively $\overline{S}_{L, M}$) as a subgraph. If $\pi$ has a realization $G$ containing $S_{L, M}$ on those vertices having degrees $d_{1}, d_{2},\ldots,d_{L+M}$, then $\pi$ is potentially $A_{L, M}$-graphic. A non-increasing sequence of non-negative integers $\pi = (d_{1}, d_{2},\ldots,d_{n})$ is potentially $A_{L, M}$-graphic if and only if it is potentially $S_{L, M}$-graphic. In this paper, we obtain the sufficient condition for a graphic sequence to be potentially $A_{L, M}$-graphic and this result is a generalization of that given by J. H. Yin on split graphs.",
issn="2251-8657",
doi="10.22108/toc.2017.20361",
url="http://toc.ui.ac.ir/article_20361.html"
}
@Article{Azadi2017,
author="Azadi, Mehrdad
and Jafari, Zeinab",
title="Some properties of comaximal ideal graph of a commutative ring",
journal="Transactions on Combinatorics",
year="2017",
volume="6",
number="1",
pages="29-37",
abstract="Let $R$ be a commutative ring with identity. We use $\varphi (R)$ to denote the comaximal ideal graph. The vertices of $\varphi (R)$ are proper ideals of R which are not contained in the Jacobson radical of $R$, and two vertices $I$ and $J$ are adjacent if and only if $I + J = R$. In this paper we show some properties of this graph together with planarity of line graph associated to $\varphi (R)$.",
issn="2251-8657",
doi="10.22108/toc.2017.20429",
url="http://toc.ui.ac.ir/article_20429.html"
}
@Article{Ariannejad2017,
author="Ariannejad, Masoud
and Emami, Mojgan
and Naserian, Ozra",
title="A family of $t$-regular self-complementary $k$-hypergraphs",
journal="Transactions on Combinatorics",
year="2017",
volume="6",
number="1",
pages="39-46",
abstract="We use the recursive method of construction large sets of t-designs given by Qiu-rong Wu (A note on extending t-designs, {\em Australas. J. Combin.}, {\bf 4} (1991) 229--235.), and present a similar method for constructing $t$-subset-regular self-complementary $k$-uniform hypergraphs of order $v$. As an application we show the existence of a new family of 2-subset-regular self-complementary 4-uniform hypergraphs with $v=16m+3$.",
issn="2251-8657",
doi="10.22108/toc.2017.20363",
url="http://toc.ui.ac.ir/article_20363.html"
}
@Article{Taghvaee2017,
author="Taghvaee, Fatemeh
and Fath-Tabar, Gholam Hossein",
title="On the skew spectral moments of graphs",
journal="Transactions on Combinatorics",
year="2017",
volume="6",
number="1",
pages="47-54",
abstract="Let $G$ be a simple graph, and $G^{\sigma}$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^{\sigma})$. The $k-$th skew spectral moment of $G^{\sigma}$, denoted by $T_k(G^{\sigma})$, is defined as $\sum_{i=1}^{n}( \lambda_{i})^{k}$, where $\lambda_{1}, \lambda_{2},\cdots, \lambda_{n}$ are the eigenvalues of $G^{\sigma}$. Suppose $G^{\sigma_1}_{1}$ and $G^{\sigma_2}_{2}$ are two digraphs. If there exists an integer $k$, $1 \leq k \leq n-1$, such that for each $i$, $0 \leq i \leq k-1$, $T_i(G^{\sigma_1}_{1}) = T_i(G^{\sigma_2}_{2})$ and $T_k(G^{\sigma_1}_{1})