On annihilator graph of a finite commutative ring
Sanghita
Dutta
North eastern Hill University
author
Chanlemki
Lanong
North Eastern Hill University
author
text
article
2017
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
1
no.
2017
1
11
http://toc.ui.ac.ir/article_20360_56c78d48b767dab5eff9143a4cf11336.pdf
dx.doi.org/10.22108/toc.2017.20360
A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs
Yun
Gao
Department of Editorial, Yunnan Normal University
author
Mohammad Reza
Farahani
Department of Applied Mathematics, Iran University of Science and Technology
author
Wei
Gao
School of Information and Technology, Yunnan Normal University
author
text
article
2017
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
1
no.
2017
13
19
http://toc.ui.ac.ir/article_20355_2293d2e8b5527d56f39b0d5e01456cad.pdf
dx.doi.org/10.22108/toc.2017.20355
The condition for a sequence to be potentially $A_{L, M}$- graphic
Shariefuddin
Pirzada
University of Kashmir
author
Bilal
A. Chat
University of Kashmir
author
text
article
2017
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
1
no.
2017
21
27
http://toc.ui.ac.ir/article_20361_5539a345ae0f45bb6974e8e9397a9145.pdf
dx.doi.org/10.22108/toc.2017.20361
Some properties of comaximal ideal graph of a commutative ring
Mehrdad
Azadi
Islamic Azad University, Central Tehran Branch
author
Zeinab
Jafari
Islamic Azad University, Central Tehran Branch
author
text
article
2017
eng
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 $\varphi (R)$.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
1
no.
2017
29
37
http://toc.ui.ac.ir/article_20429_cb19821e16c613c386c6392dde7a5d30.pdf
dx.doi.org/10.22108/toc.2017.20429
A family of $t$-regular self-complementary $k$-hypergraphs
Masoud
Ariannejad
University of zanjan
author
Mojgan
Emami
Department of Mathematics,
University of Zanjan
author
Ozra
Naserian
Department of Mathematics,
University of Zanjan
author
text
article
2017
eng
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 $v=16m+3$.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
1
no.
2017
39
46
http://toc.ui.ac.ir/article_20363_caa3ab087951b3985516a80dc389ee3a.pdf
dx.doi.org/10.22108/toc.2017.20363
On the skew spectral moments of graphs
Fatemeh
Taghvaee
University of Kashan
author
Gholam Hossein
Fath-Tabar
University of Kashan
author
text
article
2017
eng
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 $T_k(G^{\sigma_1}_{1}) <T_k(G^{\sigma_ 2}_{2})$ then we write $G^{\sigma_1}_{1} \prec_{T} G^{\sigma_2}_{2}$. In this paper, we determine some of the skew spectral moments of oriented graphs. Also we order some oriented unicyclic graphs with respect to skew spectral moment.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
1
no.
2017
47
54
http://toc.ui.ac.ir/article_20737_9c81a151b424aac06fc6253943dc89a2.pdf
dx.doi.org/10.22108/toc.2017.20737