ORIGINAL_ARTICLE
On annihilator graph of a finite commutative ring
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.
http://toc.ui.ac.ir/article_20360_56c78d48b767dab5eff9143a4cf11336.pdf
2017-03-01T11:23:20
2017-12-11T11:23:20
1
11
10.22108/toc.2017.20360
Annihilator
Clique number
Domination Number
Sanghita
Dutta
sanghita22@gmail.com
true
1
North eastern Hill University
North eastern Hill University
North eastern Hill University
LEAD_AUTHOR
Chanlemki
Lanong
lanongc@gmail.com
true
2
North Eastern Hill University
North Eastern Hill University
North Eastern Hill University
AUTHOR
[1] D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston, The zero-divisor graph of a commutative ring. II, Ideal theoretic metho ds in commutative algebra (Columbia, MO,1999), Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001 61-72.
1
[2] D. F. Anderson and P. S. Livingston, The Zero-divisor graph of a commutative ring, J. Algebra, 217 no. 2 (1999) 434-447.
2
[3] S. Akbari, H. R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r -partite graph, J. Algebra, 270 no. 1 (2003) 169-180.
3
[4] M. F. Aitiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969.
4
[5] S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J.Algebra, 274 no. 2 (2004) 847-855.
5
[6] M. Axtell, J. Stickles and W. Trampbachls, Zero-divisor ideals and realizable zero-divisor graphs, Involve, 2 no. 1 (2009) 17-27.
6
[7] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 no. 1 (2014) 108-121.
7
[8] I. Beck, Coloring of commutative rings, J. Algebra, 116 no. 1 (1998) 208-226.
8
[9] T. T. Chelvam and T. Asir, Domination in the Total Graph on Zn , Discrete Math. Algorithms Appl., 3 no. 4 (2011) 413-421.
9
[10] D. A. Mo jdeh and A. M. Rahimi, Dominating Sets of Some Graphs Asso ciated to Commutative Rings, Comm. Algebra, 40 no. 9 (2012) 3389-3396.
10
ORIGINAL_ARTICLE
A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs
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.
http://toc.ui.ac.ir/article_20355_2293d2e8b5527d56f39b0d5e01456cad.pdf
2017-03-01T11:23:20
2017-12-11T11:23:20
13
19
10.22108/toc.2017.20355
Graph
fractional factor
fractional $(k
n'
m)$-critical deleted graph
neighborhood union condition
Yun
Gao
gaoyun@ynnu.edu.cn
true
1
Department of Editorial, Yunnan Normal University
Department of Editorial, Yunnan Normal University
Department of Editorial, Yunnan Normal University
AUTHOR
Mohammad Reza
Farahani
mrfarahani88@gmail.com
true
2
Department of Applied Mathematics, Iran University of Science and Technology
Department of Applied Mathematics, Iran University of Science and Technology
Department of Applied Mathematics, Iran University of Science and Technology
AUTHOR
Wei
Gao
gaowei@ynnu.edu.cn
true
3
School of Information and Technology, Yunnan Normal University
School of Information and Technology, Yunnan Normal University
School of Information and Technology, Yunnan Normal University
LEAD_AUTHOR
[1] J. A. Bondy and U. S. R. Mutry, Graph theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008.
1
[2] W. Gao, Some results on fractional deleted graphs, Do ctoral disdertation of So o chow university, 2012.
2
[3] W. Gao and Y. Gao, Toughness condition for a graph to be a fractional (g, f, n)-critical deleted graph, The Scientic World Jo., 2014, Article ID 369798, PP. 7, http://dx.doi.org/10.1155/2014/369798.
3
[4] W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Tight toughness condition for fractional (g, f, n)-critical graphs, J. Korean Math. Soc., 51 (2014) 55-65.
4
[5] W. Gao and W. F. Wang, Toughness and fractional critical deleted graph, Util. Math., 98 (2015) 295-310.
5
[6] G. Liu and L. Zhang, Toughness and the existence of fractional k -factors of graphs, Discrete Math., 308 (2008) 1741-1748.
6
[7] J. Yu, G. Liu, M. Ma and B. Cao, A degree condition for graphs to have fractional factors, Adv. Math. (China), 35 (2006) 621-628.
7
[8] S. Z. Zhou, A minimum degree condition of fractional ( k ; m )-deleted graphs, C. R. Math. Acad. Sci., Paris, 347 (2009) 1223-1226.
8
[9] S. Z. Zhou, A neighb orho o d condition for graphs to b e fractional (k,m)-deleted graphs, Glasg. Math. J., 52 (2010) 33-40.
9
[10] S. Z. Zhou, A suﬃcient condition for a graph to b e a fractional ( f ; n )-critical graph, Glasg. Math. J. , 52 (2010) 409-415.
10
[11] S. Z. Zhou, A suﬃcient condition for graphs to b e fractional ( k ; m )-deleted graphs, Appl. Math. Lett., 24 (2011) 1533-1538.
11
[12] S. Z. Zhou and Q. Bian, An existence theorem on fractional deleted graphs, Period. Math. Hung., 71 (2015) 125-133.
12
ORIGINAL_ARTICLE
The condition for a sequence to be potentially $A_{L, M}$- graphic
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.
http://toc.ui.ac.ir/article_20361_5539a345ae0f45bb6974e8e9397a9145.pdf
2017-03-01T11:23:20
2017-12-11T11:23:20
21
27
10.22108/toc.2017.20361
Split graph
complete product split graph
potentially $H$-graphic Sequences
Shariefuddin
Pirzada
pirzadasd@kashmiruniversity.ac.in
true
1
University of Kashmir
University of Kashmir
University of Kashmir
LEAD_AUTHOR
Bilal
A. Chat
chatbilal@ymail.com
true
2
University of Kashmir
University of Kashmir
University of Kashmir
AUTHOR
[1] P. Erdos and T. Gallai, Graphs with prescrib ed degrees (in Hungarian), Matemoutiki Lapor, 11 (1960) 264-274.
1
[2] R. J. Gould, M. S. Jacobson and J. Lehel, Potentially G-graphical degree sequences, in Combinatorics, Graph Theory, and Algorithms (Y. Alavi et al., eds.), 1,2, Kalamazo o, MI, 1999 451-460.
2
[3] S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph, I. J. Soc. Indust. Appl. Math., 10 (1962) 496-506.
3
[4] V. Havel, A Remark on the existance of nite graphs (Czech), Casopis Pest. Mat., 80 (1955) 477-480.
4
[5] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient Blackswan, India, 2012.
5
[6] S. Pirzada and J. H. Yin, Degree sequences in graphs, J. Math. Study, 39 (2006) 25-31
6
[7] S. Pirzada and Bilal A. Chat, Potentially graphic sequences of split graphs, Kragujevac J. Math., 38 (2014) 73-81.
7
[8] S. Pirzada, Bilal A. Chat and Faro o q A. Dar, Graphical sequences of some family of induced subgraphs, J. Algebra Comb. Discrete Struct. Appl., 2 (2015) 95-109.
8
[9] A. R. Rao, An Erdos-Gallai type result on the clique numb er of a realization of a degree sequence, Unpublished.
9
[10] A. R. Rao, The clique number of a graph with given degree sequence, Graph Theory, Proc. Symp. Calcutta 1976, ISI Lecture Notes, 4 (A. R. Rao, ed.), 1979 251-267.
10
[11] J. H. Yin, A Havel-Hakimi typ e pro cedure and a suﬃcient condition for a sequence to be potentially Sr,s-graphic, Czechoslovak Math. J., 62 (2012) 863-867.
11
ORIGINAL_ARTICLE
Some properties of comaximal ideal graph of a commutative ring
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 $\varphi (R)$.
http://toc.ui.ac.ir/article_20429_cb19821e16c613c386c6392dde7a5d30.pdf
2017-03-01T11:23:20
2017-12-11T11:23:20
29
37
10.22108/toc.2017.20429
Comaximal graph
planar graph
line graph
Mehrdad
Azadi
meh.azadi@iauctb.ac.ir
true
1
Islamic Azad University, Central Tehran Branch
Islamic Azad University, Central Tehran Branch
Islamic Azad University, Central Tehran Branch
LEAD_AUTHOR
Zeinab
Jafari
zei.jafari.sci@iauctb.ac.ir
true
2
Islamic Azad University, Central Tehran Branch
Islamic Azad University, Central Tehran Branch
Islamic Azad University, Central Tehran Branch
AUTHOR
[1] M. Azadi, Z. Jafari and Ch. Eslahchi, On the Comaximal ideal graph of a commutative ring, Turkish J. Math., 40 (2016) 905-913.
1
[2] R. Diestel, Graph Theory, Graduate Texts in Mathematics, 173, Springer-Verlag, New York, 2000.
2
[3] F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969.
3
[4] H. R. Maimani, M. Salimi, A. Sattari and S. Yassemi, Comaximal graph of commutative rings, J. Algebra, 319 (2008) 1801-1808.
4
[5] J. Sedlacek, Some properties of interchange graphs, Theory of graphs and its applications, Academic Press, 1964 145-150.
5
[6] P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 (1995) 124-127.
6
[7] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Second edition, Cambridge University Press, Cambridge, 2001.
7
[8] W. Weisstein, Line Graph, From MathWorld-A Wolfrom WebResource.
8
[9] M. Ye and T. S. Wu, Comaximal ideal graphs of Commutative Rings, Journal of Algebra and Its Applications, 11 no. 6 (2012) pp. 14.
9
ORIGINAL_ARTICLE
A family of $t$-regular self-complementary $k$-hypergraphs
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 $v=16m+3$.
http://toc.ui.ac.ir/article_20363_caa3ab087951b3985516a80dc389ee3a.pdf
2017-03-01T11:23:20
2017-12-11T11:23:20
39
46
10.22108/toc.2017.20363
Self-complementary hypergraph
Uniform hypergraph
Regular hypergraph
Large sets of t-designs
Masoud
Ariannejad
m.ariannejad@gmail.com
true
1
University of zanjan
University of zanjan
University of zanjan
LEAD_AUTHOR
Mojgan
Emami
mojgan.emami@yahoo.com
true
2
Department of Mathematics,
University of Zanjan
Department of Mathematics,
University of Zanjan
Department of Mathematics,
University of Zanjan
AUTHOR
Ozra
Naserian
o.naserian@gmail.com
true
3
Department of Mathematics,
University of Zanjan
Department of Mathematics,
University of Zanjan
Department of Mathematics,
University of Zanjan
AUTHOR
[1] G. B. Khosrovshahi and R. Laue, t-Designs, t≤3, in: Handbook of combinatorial designs, 2nd ed. (C. J. Colb ourn and J. H. Dinitz, eds.), CRC press, Bo ca Raton, (2007) 98-110.
1
[2] G. B. Khosrovshahi, R. Laue and B. Tayfeh-Rezaie, On large sets of t-designs of size four, Bayreuth. Math. Schr., 74 (2005) 136-144.
2
[3] G. B. Khosrovshahi and B. Tayfeh-Rezaie, Ro ot cases of large sets of t-designs, Discrete Math., 263 (2003) 143-155.
3
[4] M. Knor and P. Poto cnik, A note on 2-subset-regular self-complementary 3-uniform hyp ergraphs, Ars Combin., 111 (2013) 33-36.
4
[5] W. Ko cay, Reconstructing graphs as subsumed graphs of hyp ergraphs and some self-complementary triple systems, Graphs and Combin., 8 (1992) 259-276.
5
[6] E. S. Kramer and D. M. Mesner, t-designs on hyp ergraphs, Discrete Math., 15 (1976) 263-296.
6
[7] Qiu-rong Wu, A note on extending t-designs, Australas. J. Combin., 4 (1991) 229-235.
7
[8] A. Szymanski, A note on self-complementary 4-uniform hyp ergraphs, Opuscula Math., 25 (2005) 319-323.
8
ORIGINAL_ARTICLE
On the skew spectral moments of graphs
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})
http://toc.ui.ac.ir/article_20737_9c81a151b424aac06fc6253943dc89a2.pdf
2017-03-01T11:23:20
2017-12-11T11:23:20
47
54
10.22108/toc.2017.20737
Oriented graph
skew spectral moment
skew eigenvalue
$T$-order
skew characteristic polynomial
Fatemeh
Taghvaee
taghvaei19@yahoo.com
true
1
University of Kashan
University of Kashan
University of Kashan
AUTHOR
Gholam Hossein
Fath-Tabar
fathtabar@kashanu.ac.ir
true
2
University of Kashan
University of Kashan
University of Kashan
LEAD_AUTHOR
[1] A. R. Ashra and G. H. Fath-Tabar, Bounds on the Estrada index of ISR (4,6)-fullerenes, Appl. Math. Lett., 24 (2011) 337-339.
1
[2] M. Cavers, S. M. Cioaba, S. Fallat, D. A. Gregory, W. H. Haemers, S. j. Kirkland, J. J. McDonald and M.
2
Tsatsomeros, Skew adjacency matrices of graphs, Linear Algebra Appl., 436 (2012) 5412-5429.
3
[3] X. Chen, X. Li and H. Lian, 4-Regular oriented graphs with optimum skew energy, Linear Algebra Appl., 439 (2013) 2948{2960.
4
[4] D. Cvetkovic and P. Rowlinson, Sp ectra of unicyclic graphs, Graphs Combin., 3 (1987) 7-23.
5
[5] D. Cvetkovic, M. Do ob and H. Sachs, Spectra of Graphs-Theory and Applications, 87, Academic Press, New York, 1980.
6
[6] G. H. Fath-Tabar, A. R. Ashra and I. Gutman, Note on Estrada and L Estrada indices of graphs, Bull. Cl. Sci. Math. Nat. Sci. Math., 139 (2009) 1-16.
7
[7] G. H. Fath-Tabar, A. R. Ashra and D. Stevanovic, Sp ectral Prop erties of Fullerenes, J. Comput. Theor. Nanosci., 9 (2012) 327-329.
8
[8] S. Gong and G. Xu, 3-Regular digraphs with optimum skew energy, Linear Algebra Appl., 436 (2012) 465-471.
9
[9] Y. Hou and T. Lei, Characteristic p olynomials of skew-adjacency matrices of oriented graphs, Electron. J. Combin., 18 (2011) 156-167.
10
[10] B. Shader and W. So, Skew sp ectra of oriented graphs, Electron. J. Combin., 16 (2009) 1-6.
11
[11] F. Taghvaee and A. R. Ashra, Comparing fullerenes by sp ectral moments, J. Nanosci. Nanotechnol., 16 (2016) 1-4.
12
[12] F. Taghvaee and G. H. Fath-Tabar, Signless Laplacian sp ectral moments of graphs and ordering some graphs with resp ect to them, Alg. Struc. Appl., 1 (2014) 133-141.
13
[13] F. Taghvaee and G. H. Fath-Tabar, Relationship b etween co eﬃcients of characteristic p olynomial and matching p olynomial of regular graphs and its applications, Iranian J. Math. Chem., 8 (2017) 7-24.
14
[14] Y. P. Wu and H. Q. Liu, Lexicographical ordering by sp ectral moments of trees with a prescrib ed diameter, Linear Algebra Appl., 433 (2010) 1707-1713.
15