2017
6
1
1
0
On annihilator graph of a finite commutative ring
2
2
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.
1

1
11


Sanghita
Dutta
North eastern Hill University
North eastern Hill University
India
sanghita22@gmail.com


Chanlemki
Lanong
North Eastern Hill University
North Eastern Hill University
India
lanongc@gmail.com
Annihilator
Clique number
Domination Number
A neighborhood union condition for fractional $(k,n',m)$critical deleted graphs
2
2
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 $kge 2$, $n',mge0$, $nge8k+n'+4m7$, and $delta(G)ge k+n'+m$, if $$N_{G}(x)cup N_{G}(y)gefrac{n+n'}{2}$$ for each pair of nonadjacent 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.
1

13
19


Yun
Gao
Department of Editorial, Yunnan Normal University
Department of Editorial, Yunnan Normal University
China
gaoyun@ynnu.edu.cn


Mohammad Reza
Farahani
Department of Applied Mathematics, Iran University of Science and Technology
Department of Applied Mathematics, Iran University
China
mrfarahani88@gmail.com


Wei
Gao
School of Information and Technology, Yunnan Normal University
School of Information and Technology, Yunnan
China
gaowei@ynnu.edu.cn
Graph
fractional factor
fractional $(k
n'
m)$critical deleted graph
neighborhood union condition
The condition for a sequence to be potentially $A_{L, M}$ graphic
2
2
The set of all nonincreasing nonnegative integer sequences $pi=(d_1, d_2,ldots,d_n)$ is denoted by $NS_n$. A sequence $piin NS_{n}$ 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 $L = sumlimits_{i = 1}^{p}r_{i}$ and $M = sumlimits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers. Another split graph is denoted by $S_{L, M} = overline{S}_{r_{1}, s_{1}} veeoverline{S}_{r_{2}, s_{2}} vee cdots vee overline{S}_{r_{p}, s_{p}}= (K_{r_{1}} vee overline{K}_{s_{1}})vee (K_{r_{2}} vee overline{K}_{s_{2}})vee cdots vee (K_{r_{p}} vee overline{K}_{s_{p}})$. 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 nonincreasing sequence of nonnegative 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.
1

21
27


Shariefuddin
Pirzada
University of Kashmir
University of Kashmir
India
pirzadasd@kashmiruniversity.ac.in


Bilal
A. Chat
University of Kashmir
University of Kashmir
India
chatbilal@ymail.com
Split graph
complete product split graph
potentially $H$graphic Sequences
Some properties of comaximal ideal graph of a commutative ring
2
2
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 $varphi (R)$.
1

29
37


Zeinab
Jafari
Islamic Azad University, Central Tehran Branch
Islamic Azad University, Central Tehran Branch
Iran
zei.jafari.sci@iauctb.ac.ir


Mehrdad
Azadi
Islamic Azad University, Central Tehran Branch
Islamic Azad University, Central Tehran Branch
Iran
meh.azadi@iauctb.ac.ir
Comaximal graph
planar graph
line graph
A family of $t$regular selfcomplementary $k$hypergraphs
2
2
We use the recursive method of construction large sets of tdesigns given by Qiurong Wu (A note on extending tdesigns, {em Australas. J. Combin.}, {bf 4} (1991) 229235.), and present a similar method for constructing $t$subsetregular selfcomplementary $k$uniform hypergraphs of order $v$. As an application we show the existence of a new family of 2subsetregular selfcomplementary 4uniform hypergraphs with $v=16m+3$.
1

39
46


Masoud
Ariannejad
University of zanjan
University of zanjan
Iran
m.ariannejad@gmail.com


Mojgan
Emami
Department of Mathematics,
University of Zanjan
Department of Mathematics,
University of
Iran
mojgan.emami@yahoo.com


Ozra
Naserian
Department of Mathematics,
University of Zanjan
Department of Mathematics,
University of
Iran
o.naserian@gmail.com
Selfcomplementary hypergraph
Uniform hypergraph
Regular hypergraph
Large sets of tdesigns
On the skew spectral moments of graphs
2
2
Let $G$ be a simple graph, and $G^{sigma}$ be an oriented graph of $G$ with the orientation $sigma$ and skewadjacency 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 $k$, $1 leq k leq n1$, such that for each $i$, $0 leq i leq k1$, $T_i(G^{sigma_1}_{1}) = T_i(G^{sigma_2}_{2})$ 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.
1

47
54


Gholam Hossein
FathTabar
University of Kashan
University of Kashan
Iran
fathtabar@kashanu.ac.ir


Fatemeh
Taghvaee
University of Kashan
University of Kashan
Iran
taghvaei19@yahoo.com
Oriented graph
skew spectral moment
skew eigenvalue
$T$order
skew characteristic polynomial