2018
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3
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73
The annihilator graph of a 0distributive lattice
2
2
In this article, for a lattice $mathcal L$, we define and investigate the annihilator graph $mathfrak {ag} (mathcal L)$ of $mathcal L$ which contains the zerodivisor graph of $mathcal L$ as a subgraph. Also, for a 0distributive lattice $mathcal L$, we study some properties of this graph such as regularity, connectedness, the diameter, the girth and its domination number. Moreover, for a distributive lattice $mathcal L$ with $Z(mathcal L)neqlbrace 0rbrace$, we show that $mathfrak {ag} (mathcal L) = Gamma(mathcal L)$ if and only if $mathcal L$ has exactly two minimal prime ideals. Among other things, we consider the annihilator graph $mathfrak {ag} (mathcal L)$ of the lattice $mathcal L=(mathcal D(n),)$ containing all positive divisors of a nonprime natural number $n$ and we compute some invariants such as the domination number, the clique number and the chromatic number of this graph. Also, for this lattice we investigate some special cases in which $mathfrak {ag} (mathcal D(n))$ or $Gamma(mathcal D(n))$ are planar, Eulerian or Hamiltonian.
1

1
18


Saeid
Bagheri
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran
Department of Mathematics, Faculty of Mathematical
Iran
bagheri_saeid@yahoo.com


Mahtab
Koohi Kerahroodi
Department of Mathematics, Faculty of Mathematical Sciences, Malayer University, Malayer, Iran.
Department of Mathematics, Faculty of Mathematical
Iran
mahtabkh3@gmail.com
Distributive lattice
Annihilator graph
Zerodivisor graph
[[1] M. Afkhami, K. Khashyarmanesh and Z. Rajabi, Some results on the annihilator graph of a commutative ring, Czechoslovak Math. Journal, 67 (2017) 151–169. ##[2] S. Akbari and A. Mohammadian, On zerodivisor graph of finite rings, J. Algebra, 314 (2007) 168–184. ##[3] M. Alizadeh, A. K. Das, H. R. Maimani, M. R. Pournaki and S. Yassemi, On the diameter and girth of zerodivisor graphs of posets, Discrete Appl. Math., 160 (2012) 1319–1324. ##[4] D. D. Anderson and M. Naseer, Beck‘s coloring of a commutative ring, J. Algebra, 159 (1993) 500514. ##[5] D. F. Anderson and P. Livingston, The zerodivisor graph of a commutative ring, J. Algebra, 217 (1999) 434447. ##[6] D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008) 2706–2719. ##[7] D. F. Anderson and S. B. Mulay, On the diameter and girth of a zerodivisor graph, J. Pure Appl. Algebra, 210 (2007) 543–550. ##[8] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 (2014) 108–121. ##[9] I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988) 208–226. ##[10] B. Bollobas and I. Rival, The maximal size of the covering graph of a lattice, Algebra Univ., 9 (1979) 371–373. ##[11] J. Coykendal, S. SatherWagstaff, L. Sheppardson and S. Spiroff, On zero divisor graphs, Progress in commutative Algebra, 2 (2012) 241–299. ##[12] F. R. Demeyer, T. Mckenzie and K. Schneider, The zero divisor graph of a commutative semigroup, Semigroup Forum, 65 (2002) 206–214. ##[13] D. Duffus and I. Rival, Path length in the covering graph of a lattice, Discrete Math., 19 (1977) 139–158. ##[14] S. Dutta and Ch. Lanong, On annihilator graphs of a finite commutative ring, Trans. Comb., 6 no. 1 (2017) 111. ##[15] E. Estaji and K. Khashyarmanesh, The zero divisor graph of a lattice, Results Math., 61 (2012) 1–11. ##[16] N. D. Filipov, Comparability graphs of partially ordered sets of different types, Colloq. Math. Soc. Janos Bolyai, 33 (1980) 373–380. ##[17] E. Gedeonova, Lattices whose covering graphs are Sgraphs, Colloq. Math. Soc. Janos Bolyai, 33 (1980) 407435. ##[18] G. Grätzer, Lattice Theory: Foundation, Birkhauser, Basel, 2011. ##[19] V. Joshi, Zero divisor graphs of a poset with respect to an ideal, Order, 29 (2012) 499–506. ##[20] V. Joshi and A. Khiste, On the zero divisor graphs of pmlattices, Discrete Math., 312 (2012) 2076–2082. ##[21] V. Joshi and S. Sarode, Diameter and girth of zero divisor graph of multiplicative lattices, AsianEur. J. Math., 9 (2016). http://dx.doi.org/10.1142/S1793557116500716. ##[22] T. G. Lucas, The diameter of a zero divisor graph, J. Algebra, 301 (2006) 174–193. ##[23] M. J. Nikmehr, R. Nikandish and M. Bakhtyiari, More on the annihilator graph of a commutative ring, Hokkaido Math. J., 46 (2017) 107–118. ##[24] Y. S. Pawar and N. K. Thakare, pmlattices, Algebra Univ., 7 (1977) 259–263. ##[25] T. Tamizh Chelvam and S. Nithya, A note on the zero divisor graph of a lattice, Trans. Comb., 3 no. 3 (2014) 51–59. ##[26] D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall Upper Saddle River, 2001. ##[27] R. J. Wilson, Introduction to Graph Theory, Fourth edition, Longman, Harlow, 1996.##]
A spectral excess theorem for digraphs with normal Laplacian matrices
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2
The spectral excess theorem, due to Fiol and Garriga in 1997, is an important result, because it gives a good characterization of distanceregularity in graphs. Up to now, some authors have given some variations of this theorem. Motivated by this, we give the corresponding result by using the Laplacian spectrum for digraphs. We also illustrate this Laplacian spectral excess theorem for digraphs with few Laplacian eigenvalues and we show that any strongly connected and regular digraph that has normal Laplacian matrix with three distinct eigenvalues, is distanceregular. Hence such a digraph is strongly regular with girth $g=2$ or $g=3$.
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19
28


Fateme
Shafiei
Isfahan University of Technology
Isfahan University of Technology
Iran
fatemeh.shafiei66@gmail.com
A Laplacian spectral excess theorem
Distanceregular digraphs
Strongly regular digraphs
[[1] J. BangJensen and G. Z. Gregory, Digraphs: Theory, Algorithms and Applications, Springer Monographs in##Mathematics, 2nd ed, 2009. ##[2] A. E. Brouwer, personal homepage: http://www.cwi.nl/~aeb/math/dsrg/dsrg.html. ##[3] A. E. Brouwer, A. M. Cohen and A. Neumaier, distanceregular Graphs, SpringerVerlag, BerlinNew York,##[4] A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Springer, 2012; available online at http://homepages.cwi.nl/~aeb/math/ipm/. ##[5] F. Comellas, M. A. Fiol, J. Gimbert and M. Mitjana, Weakly distanceregular digraphs, J. Combin. Theory##Ser. B, 90 (2004) 233–255. ##[6] D. M. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs, Theory and Application, VEB Deutscher Verlag##der Wissenschaften, Berlin, second edition, 1982. ##[7] C. Dalfo, E.R. Van Dam, M. A. Fiol and E. Garriga, Dual concepts of almost distanceregularity and the##spectral excess theorem, Discrete Math., 312 (2012) 2730–2734. ##[8] C. Dalfo, E. R. Van Dam, M. A. Fiol, E. Garriga and B. L. Gorissen, On almost distanceregular graphs, J.##Combin. Theory Ser. A, 118 (2011) 1094–1113. ##[9] R. M. Damerell, Distancetransitive and distanceregular digraphs, J. Combin. Theory Ser. B, 31 (1981) 46–53. ##[10] A. M. Duval, A directed graph version of strongly regular graphs, J. Combin. Theory Ser. A, 47 (1988) 71–100. ##[11] E. R. Van Dam, The spectral excess theorem for distanceregular graphs: a global (over)view, Electron. J.##Combin., 15 (2008), pp. 10. ##[12] E. R. Van Dam and M. A. Fiol, A short proof of the oddgirth theorem, Electron. J. Combin., 19 (2012) pp.##[13] E. R. Van Dam and M. A. Fiol, The Laplacian Spectral Excess Theprem for distanceRegular Graphs, Linear##Algebra Appl., 458 (2014) 1–6. ##[14] Carl D. Meyer, Matrix analysis and applied linear algebra, Philadelphia, USA , 101 (2011) 486–489. ##[15] M. A. Fiol, Algebraic characterizations of distanceregular graphs, Discrete Math., 246 (2002) 111–129. ##[16] M. A. Fiol, On some approaches to the spectral excess theorem for nonregular graphs, J. Combin. Theory Ser. A, 120 (2013) 1285–1290. ##[17] M. A. Fiol and E. Garriga, From local adjacency polynomials to locally pseudodistanceregular graphs, J.##Combin. Theory Ser. B, 71 (1997) 162–183. ##[18] M. A. Fiol, S. Gago and E. Garriga, A simple proof of the spectral excess theorem for distanceregular graphs, Linear Algebra Appl., 432 (2010) 2418–2422. ##[19] M. A. Fiol, E. Garriga and J. L. A. Yebra, Locally pseudodistanceregular graphs, J. Combin.Theory Ser. B,##68 (1996) 179–205. ##[20] G. S. Lee and C. W. Weng, The spectral excess theorem for general graphs, J. Combin. Theory Ser. A, 119 (2012) 1427–1431. ##[21] G. R. Omidi, A spectral excess theorem for normal digraphs, J. Algebraic Combin., 42 (2015) 537–554.##]
Sufficient conditions for trianglefree graphs to be super$λ'$
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2
An edgecut $F$ of a connected graph $G$ is called a restricted edgecut if $GF$ contains no isolated vertices. The minimum cardinality of all restricted edgecuts is called the restricted edgeconnectivity $λ'(G)$ of $G$. A graph $G$ is said to be $λ'$optimal if $λ'(G)=xi(G)$, where $xi(G)$ is the minimum edgedegree of $G$. A graph is said to be super$λ'$ if every minimum restricted edgecut isolates an edge. In this paper, first, we provide a short proof of a previous theorem about the sufficient condition for $λ'$optimality in trianglefree graphs, which was given in [J. Yuan and A. Liu, Sufficient conditions for $λ_k$optimality in trianglefree graphs, Discrete Math., 310 (2010) 981987]. Second, we generalize a known result about the sufficient condition for trianglefree graphs being super$λ'$ which was given by Shang et al. in [L. Shang and H. P. Zhang, Sufficient conditions for graphs to be $λ'$optimal and super$λ'$, Network}, 309 (2009) 33363345].
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29
36


Huiwen
Cheng
Department of Mathematics, Zhongguancun Institute
Department of Mathematics, Zhongguancun
China
15275013@qq.com


YanJing
Li
Department of Mathematics, Beijing Jiaotong University, China
Department of Mathematics, Beijing Jiaotong
China
05118322@bjtu.edu.cn
Trianglefree
restricted edgecut
super$ld'$
$mathcal{B}$Partitions, determinant and permanent of graphs
2
2
Let $G$ be a graph (directed or undirected) having $k$ number of blocks $B_1, B_2,hdots,B_k$. A $mathcal{B}$partition of $G$ is a partition consists of $k$ vertexdisjoint subgraph $(hat{B_1},hat{B_1},hdots,hat{B_k})$ such that $hat{B}_i$ is an induced subgraph of $B_i$ for $i=1,2,hdots,k.$ The terms $prod_{i=1}^{k}det(hat{B}_i), prod_{i=1}^{k}text{per}(hat{B}_i)$ represent the detsummands and the persummands, respectively, corresponding to the $mathcal{B}$partition $(hat{B_1},hat{B_1},hdots,hat{B_k})$. The determinant (permanent) of a graph having no loops on its cutvertices is equal to the summation of the detsummands (persummands), corresponding to all possible $mathcal{B}$partitions. In this paper, we calculate the determinant and the permanent of classes of graphs such as block graph, block graph with negatives cliques, signed unicyclic graph, mixed complete graph, negative mixed complete graph, and star mixed block graphs.
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37
54


Ranveer
Singh
Department of Mathematics, Indian Institute of Technology Jodhpur, Jodhpur, India
Department of Mathematics, Indian Institute
India
pg201283008@iitj.ac.in


Ravindra
Bapat
StatMath Unit, ISI Delhi
StatMath Unit, ISI Delhi
India
rbb@isid.ac.in
$mathcal{B}$partition
signed graph
mixed block graph
[[1] B. D. Acharya, Spectral criterion for cycle balance in networks, J. Graph Theory, 4 (1980) 1–11.##[2] R. B. Bapat, Graphs and matrices, 27, Springer, London; Hindustan Book Agency, New Delhi, 2010.##[3] R. B. Bapat and S. Roy, On the adjacency matrix of a block graph, Linear and Multilinear Algebra, 62 (2014) 406–418.##[4] D. Cartwright and F. Harary, Structural balance: a generalization of Heider’s theory, Psychol Rev., 63 (1956) 277–293.##[5] J. Ding and A. Zhou, Eigenvalues of rankone updated matrices with some applications, Appl. Math. Lett., 20 (2007)##1223–1226.##[6] D. Easley and J. Kleinberg, Networks, crowds, and markets, 6, Cambridge Univ Press, 2010.##[7] F. Harary, On the notion of balance of a signed graph, Michigan Math. J., 2 (1953–54) 143–146.##[8] T. Amdeberhan, Determinant of a matrix having diagonal and subdiagonal entries zero, MathOverflow, http://##mathoverflow.net/q/264167, (version: 20170310).##[9] R. Singh, Determinant of a matrix having diagonal and subdiagonal entries zero, MathOverflow, http://mathoverflow.##net/q/264264, (version: 20170310).##[10] R. Singh and R. B. Bapat, Eigenvalues of some signed graphs with negative cliques, arXivpreprintarXiv:1702.06322,##[11] R. Singh and R. B. Bapat, On characteristic and permanent polynomials of a matrix, Spec. Matrices, 5 (2017) 97–112.##[12] H. Zhou, The inverse of the distance matrix of a distance welldefined graph, Linear Algebra Appl., 517 (2017) 11–29.##]
Iota energy of weighted digraphs
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2
The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. The iota energy of a digraph is recently defined as the sum of absolute values of imaginary part of its eigenvalues. In this paper, we extend the concept of iota energy of digraphs to weighted digraphs. We compute the iota energy formulae for the positive and negative weight directed cycles. We also characterize the unicyclic weighted digraphs with cycle weight $ r in [1, 1]backslash {0}$ having minimum and maximum iota energy. We obtain well known McClelland upper bound for the iota energy of weighted digraphs. Finally, we find the class of noncospectral equienergetic weighted digraphs.
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55
73


Sumaira
Hafeez
School of Natural Sciences, National university of sciences and Technology Islamabad, Pakistan
School of Natural Sciences, National university
Pakistan
sumaira.hafeez123@gmail.com


Mehtab
Khan
Department of mathematics, school of Natural Sciences, National University of Sciences and Technology Islamabad, Pakistan
Department of mathematics, school of Natural
Pakistan
mehtabkhan85@gmail.com
Weighted digraphs
Extremal energy
Equienergetic weighted digraphs