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106 results on '"incidence graph"'

1. A topological property of a hypergraph assigned to commutative rings.

Author

Selvakumar, K., Beautlin Jemi, J., and Moniri Hamzekolaee, A. R.

Subjects
*COMMUTATIVE rings, *MOLECULAR connectivity index, *TOPOLOGICAL property
Abstract

Studying algebraic structures via graphs and hypergraphs assigned to them can be of interest. Especially, computing the genus of a graph as a topological index leads to a better understanding of the related algebraic structure. In this direction we apply a hypergraph, namely 3 -zero divisor hypergraph assigned to a commutative ring and study its genus. In this paper, we characterize all finite commutative nonlocal rings A with identity whose ℋ 3 (A) has genus two. Further, we classify all finite commutative nonlocal rings A whose ℋ 3 (A) has crosscap two. Moreover, we provide a MATLAB code for calculating 3 -zero-divisor of ℤ n and the hyperedge of 3 -zero-divisor hypergraph of  ℤ n . [ABSTRACT FROM AUTHOR]

Published
2025
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2. Characterizing Bipartite Distance-Regularized Graphs with Vertices of Eccentricity 4.

Author

Fernández, Blas, Maksimović, Marija, and Rukavina, Sanja

Abstract

Consider a bipartite distance-regularized graph Γ with color partitions Y and Y ′ . Notably, all vertices in partition Y (and similarly in Y ′ ) exhibit a shared eccentricity denoted as D (and D ′ , respectively). The characterization of bipartite distance-regularized graphs, specifically those with D ≤ 3 , in relation to the incidence structures they represent is well established. However, when D = 4 , there are only two possible scenarios: either D ′ = 3 or D ′ = 4 . The instance where D = 4 and D ′ = 3 has been previously investigated. In this paper, we establish a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters (v , b , r , k , λ 1 , 0) of type (k - 1 , t) , featuring intersection numbers x = 0 and y > 0 (where y ≤ t < k ), and bipartite distance-regularized graphs with D = D ′ = 4 . Moreover, our investigations result in the systematic classification of 2-Y-homogeneous bipartite distance-regularized graphs, which are incidence graphs of quasi-symmetric SPBIBDs with parameters (v , b , r , k , λ 1 , 0) of type (k - 1 , t) with intersection numbers x = 0 and y = 1 . [ABSTRACT FROM AUTHOR]

Published
2024
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3. The projective 3-annihilating-ideal hypergraphs.

Author

Ramanathan, V. and Selvaraj, C.

Subjects
*COMMUTATIVE rings, *INTERSECTION graph theory, *HYPERGRAPHS, *ISOMORPHISM (Mathematics)
Abstract

In this paper, we investigate the crosscap of 3-annihilating-ideal hypergraph 3 (R) of a commutative ring R and the topological embedding of 3 (R) to the nonorientable compact surfaces. Furthermore, we determine all Artinian commutative non-local rings R (up to isomorphism) such that 3 (R) is a projective graph. [ABSTRACT FROM AUTHOR]

Published
2022
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4. The non-orientable 4-genus for knots with 10 crossings.

Subjects
*KNOT theory, *MINIMAL surfaces, *BETTI numbers
Abstract

Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10. [ABSTRACT FROM AUTHOR]

Published
2022
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5. Monotonic Normalized Heat Diffusion for Regular Bipartite Graphs with Four Eigenvalues.

Author

Kubo, Tasuku and Namba, Ryuya

Abstract

Let X = (V , E) be a finite regular graph and H t (u , v) , u , v ∈ V , the heat kernel on X. We prove that, if the graph X is bipartite and has four distinct Laplacian eigenvalues, the ratio H t (u , v) / H t (u , u) , u , v ∈ V , is monotonically non-decreasing as a function of t. The key to the proof is the fact that such a graph is an incidence graph of a symmetric 2-design. [ABSTRACT FROM AUTHOR]

Published
2022
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6. The -annihilating-ideal hypergraph of commutative ring

Author

K. Selvakumar and V. Ramanathan

Subjects
annihilating-ideal, hypergraph, incidence graph, planar, -annihilating-ideal hypergraph, Mathematics, QA1-939
Abstract

The concept of the annihilating-ideal graph of a commutative ring was introduced by Behboodi et. al in 2011. In this paper, we extend this concept to the hypergraph for which we define an algebraic structure called -annihilating-ideal of a commutative ring which is the vertex set of the hypergraph of such commutative ring. Let be a commutative ring and an integer greater than 2 and let be the set of all -annihilating-ideals of . The -annihilating-ideal hypergraph of , denoted by , is a hypergraph with vertex set , and for distinct elements in , the set is an edge of if and only if and the product of any elements of the is nonzero. In this paper, we provide a necessary and sufficient condition for the completeness of 3-annihilating-ideal hypergraph of a commutative ring. Further, we study the planarity of and characterize all commutative ring whose 3-annihilating-ideal hypergraph is planar.

Published
2019
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8. An Upper Bound for Resolution Size: Characterization of Tractable SAT Instances

Author

Imanishi, Kensuke, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Poon, Sheung-Hung, editor, Rahman, Md. Saidur, editor, and Yen, Hsu-Chun, editor

Published
2017
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9. Performance Analysis of Expander Graph Based Key Predistribution Scheme in WSN

Author

Saikia, Monjul, Hussain, Md. A., Filipe, Joaquim, Editorial Board Member, Ghosh, Ashish, Editorial Board Member, Kotenko, Igor, Editorial Board Member, Prates, Raquel Oliveira, Editorial Board Member, Zhou, Lizhu, Editorial Board Member, Barbosa, Simone Diniz Junqueira, Editorial Board Member, Chen, Phoebe, Editorial Board Member, Du, Xiaoyong, Editorial Board Member, Kara, Orhun, Editorial Board Member, Liu, Ting, Editorial Board Member, Sivalingam, Krishna M., Editorial Board Member, Washio, Takashi, Editorial Board Member, Yuan, Junsong, Editorial Board Member, Unal, Aynur, editor, Nayak, Malaya, editor, Mishra, Durgesh Kumar, editor, Singh, Dharm, editor, and Joshi, Amit, editor

Published
2016
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10. Generalized polygons

Author

De Bruyn, Bart, Bunimovich, Leonid, Advisory editor, Chen, William Y.C., Advisory editor, Perthame, Benoît, Advisory editor, Saloff-Coste, Laurent, Advisory editor, Shparlinski, Igor, Advisory editor, Sprößig, Wolfgang, Advisory editor, Villani, Cédric, Advisory editor, and De Bruyn, Bart

Published
2016
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11. Algorithmic Applications of Tree-Cut Width

Author

Ganian, Robert, Kim, Eun Jung, Szeider, Stefan, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Italiano, Giuseppe F., editor, Pighizzini, Giovanni, editor, and Sannella, Donald T., editor

Published
2015
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12. On Planar Boolean CSP

Author

Dvořák, Zdeněk, Kupec, Martin, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Halldórsson, Magnús M., editor, Iwama, Kazuo, editor, Kobayashi, Naoki, editor, and Speckmann, Bettina, editor

Published
2015
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13. Specular Sets

Author

Berthé, Valérie, De Felice, Clelia, Delecroix, Vincent, Dolce, Francesco, Leroy, Julien, Perrin, Dominique, Reutenauer, Christophe, Rindone, Giuseppina, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Manea, Florin, editor, and Nowotka, Dirk, editor

Published
2015
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14. On the genus of the k-annihilating-ideal hypergraph of commutative rings.

Author

Selvakumar, K. and Ramanathan, V.

Abstract

Let R be a commutative ring and k an integer greater than 2 and let A (R , k) be the set of all k-annihilating-ideals of R. The k-annihilating-ideal hypergraph of R, denoted by A G k (R) , is a hypergraph with vertex set A (R , k) , and for distinct elements I1, ..., Ik in A (R , k) , the set {I1, I2, ..., Ik} is an edge of A G k (R) if and only if ∏ i = 1 k I i = (0) and the product of any (k − 1) elements of the set {I1, I2,..., Ik} is nonzero. In this paper, we characterize all Artinian commutative nonlocal rings R whose A G 3 (R) has genus one. [ABSTRACT FROM AUTHOR]

Published
2019
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15. The vertex-isoperimetric number of the incidence and non-incidence graphs of unitals.

Author

Hui, Alice M. W., Surani, Muhammad Adib, and Zhou, Sanming

Subjects
MAGNITUDE (Mathematics)
Abstract

We derive upper and lower bounds for the vertex-isoperimetric number of the incidence graphs of unitals and determine its order of magnitude. In the case when a unital contains sufficiently large arcs, these bounds agree and give rise to the precise value of this parameter. In particular, we obtain the exact value of the vertex-isoperimetric number of the incidence graphs of classical unitals and a certain subfamily of BM-unitals. In the case when the maximum size of arcs in the unital is relatively small, we obtain an upper bound for this parameter in terms of the vertex-isoperimetric number of the incidence graph. We also determine the exact value of the vertex-isoperimetric number of the non-incidence graph of any unital. [ABSTRACT FROM AUTHOR]

Published
2019
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16. Sankaku-Tori: An Old Western-Japanese Game Played on a Point Set

Author

Horiyama, Takashi, Kiyomi, Masashi, Okamoto, Yoshio, Uehara, Ryuhei, Uno, Takeaki, Uno, Yushi, Yamauchi, Yukiko, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Kobsa, Alfred, editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Weikum, Gerhard, editor, Ferro, Alfredo, editor, Luccio, Fabrizio, editor, and Widmayer, Peter, editor

Published
2014
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17. Graph of Ultrametric Type Matrices

Author

Dellacherie, Claude, Martinez, Servet, San Martin, Jaime, Morel, J.-M., Editor-in-chief, Teissier, Bernard, Editor-in-chief, De Lellis, Camillo, Series editor, di Bernardo, Mario, Series editor, Figalli, Alessio, Series editor, Khoshnevisan, Davar, Series editor, Kontoyiannis, Ioannis, Series editor, Lugosi, Gabor, Series editor, Podolskij, Mark, Series editor, Serfaty, Sylvia, Series editor, Stroppel, Catharina, Series editor, Wienhard, Anna, Series editor, Dellacherie, Claude, Martinez, Servet, and San Martin, Jaime

Published
2014
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21. MSOL Restricted Contractibility to Planar Graphs

Author

Abello, James, Klavík, Pavel, Kratochvíl, Jan, Vyskočil, Tomáš, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Thilikos, Dimitrios M., editor, and Woeginger, Gerhard J., editor

Published
2012
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22. Backdoors to Satisfaction

Author

Gaspers, Serge, Szeider, Stefan, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Bodlaender, Hans L., editor, Downey, Rod, editor, Fomin, Fedor V., editor, and Marx, Dániel, editor

Published
2012
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24. On the planarity of the k-zero-divisor hypergraphs

Author

T. Tamizh Chelvam, K. Selvakumar, and V. Ramanathan

Subjects
Hypergraph, Zero-divisor graph, Planar hypergraph, Incidence graph, Mathematics, QA1-939
Abstract

Let R be a commutative ring with identity and let Z(R,k) be the set of all k-zero-divisors in R and k>2 an integer. The k-zero-divisor hypergraph of R, denoted by Hk(R), is a hypergraph with vertex set Z(R,k), and for distinct element x1,x2,…,xk in Z(R,k), the set {x1,x2,…,xk} is an edge of Hk(R) if and only if x1x2⋯xk=0 and the product of elements of no (k−1)-subset of {x1,x2,…,xk} is zero. In this paper, we characterize all finite commutative non-local rings R for which the k-zero-divisor hypergraph is planar.

Published
2015
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25. Inclusion/Exclusion Branching for Partial Dominating Set and Set Splitting

Author

Nederlof, Jesper, van Rooij, Johan M. M., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Raman, Venkatesh, editor, and Saurabh, Saket, editor

Published
2010
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26. On the Boolean-Width of a Graph: Structure and Applications

Author

Adler, Isolde, Bui-Xuan, Binh-Minh, Rabinovich, Yuri, Renault, Gabriel, Telle, Jan Arne, Vatshelle, Martin, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, and Thilikos, Dimitrios M., editor

Published
2010
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28. Edge-primitive Cayley graphs on abelian groups and dihedral groups.

Author

Pan, Jiangmin, Wu, Cixuan, and Yin, Fugang

Subjects
*CAYLEY graphs, *ABELIAN groups, *DIHEDRAL angles, *EDGES (Geometry), *GEOMETRIC vertices
Abstract

Abstract A graph is called edge-primitive if its automorphism group acts primitively on its edge set. In 1973, Weiss (1973) determined all edge-primitive graphs of valency three, and recently Guo et al. (2013,2015) classified edge-primitive graphs of valencies four and five. In this paper, we determine all edge-primitive Cayley graphs on abelian groups and dihedral groups. [ABSTRACT FROM AUTHOR]

Published
2018
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29. Hamiltonian Property of the Incidence Graphs of Quadrangles Associated with Symplectic Forms on Finite Fields.

Author

Sato, Hajime and Suzuki, Hiroshi

Subjects
*HAMILTONIAN graph theory, *HAMILTONIAN mechanics, *MATHEMATICAL analysis, *MATHEMATICS theorems, *GRAPH theory
Abstract

We show that the incidence graphs of finite generalized quadrangles associated with symplectic forms on finite fields are Hamiltonian. This is an extension of Singer’s theorem (Singer Trans Am Math Soc 43:377-385, 1938) on generalized triangles to certain classical polygons. [ABSTRACT FROM AUTHOR]

Published
2018
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30. The Terwilliger algebra of the incidence graph of the Hamming graph.

Author

Tomiyama, Masato

Abstract

Let Γ be a distance-semiregular graph on Y, and let DY be the diameter of Γ on Y. Let Δ be the halved graph of Γ on Y. Fix x∈Y. Let T and T′ be the Terwilliger algebras of Γ and Δ with respect to x, respectively. Assume, for an integer i with 1≤2i≤DY and for y,z∈Γ2i(x) with ∂Γ(y,z)=2, the numbers |Γ2i-1(x)∩Γ(y)∩Γ(z)| and |Γ2i+1(x)∩Γ(y)∩Γ(z)| depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of Γ and T′-modules of Δ. Assume Γ is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set C of all maximal cliques. Then, Γ satisfies above assumption and Δ is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of Γ. For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W. [ABSTRACT FROM AUTHOR]

Published
2018
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31. On the metric dimension of incidence graphs.

Author

Bailey, Robert F.

Subjects
*REGULAR graphs, *DIMENSIONS, *INCIDENCE functions, *MATHEMATICAL symmetry, *MATHEMATICAL bounds
Abstract

A resolving set for a graph Γ is a collection of vertices S , chosen so that for each vertex v , the list of distances from v to the members of S uniquely specifies v . The metric dimension μ ( Γ ) is the smallest size of a resolving set for Γ . We consider the metric dimension of two families of incidence graphs: incidence graphs of symmetric designs, and incidence graphs of symmetric transversal designs (i.e. symmetric nets). These graphs are the bipartite distance-regular graphs of diameter 3, and the bipartite, antipodal distance-regular graphs of diameter 4, respectively. In each case, we use the probabilistic method in the manner used by Babai to obtain bounds on the metric dimension of strongly regular graphs, and are able to show that μ ( Γ ) = O ( n log n ) (where n is the number of vertices). [ABSTRACT FROM AUTHOR]

Published
2018
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33. The Computational Complexity of the Minimum Weight Processor Assignment Problem

Author

Broersma, Hajo J., Paulusma, Daniel, Smit, Gerard J. M., Vlaardingerbroek, Frank, Woeginger, Gerhard J., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Hromkovič, Juraj, editor, Nagl, Manfred, editor, and Westfechtel, Bernhard, editor

Published
2005
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34. Recognizable Sets of Graphs, Hypergraphs and Relational Structures: A Survey

Author

Courcelle, Bruno, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Calude, Cristian S., editor, Calude, Elena, editor, and Dinneen, Michael J., editor

Published
2005
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35. Equivalence Between Regular n-G-Maps and n-Surfaces

Author

Alayrangues, Sylvie, Daragon, Xavier, Lachaud, Jacques-Olivier, Lienhardt, Pascal, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Klette, Reinhard, editor, and Žunić, Joviša, editor

Published
2005
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37. On Finding Short Resolution Refutations and Small Unsatisfiable Subsets

Author

Fellows, Michael R., Szeider, Stefan, Wrightson, Graham, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Downey, Rod, editor, Fellows, Michael, editor, and Dehne, Frank, editor

Published
2004
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40. Classification of non-local rings with projective 3-zero-divisor hypergraph.

Author

Selvakumar, K. and Ramanathan, V.

Abstract

Let R be a commutative ring with identity and let Z( R, k) be the set of all k-zero-divisors in R and $$k>2$$ an integer. The k- zero-divisor hypergraph of R, denoted by $$\mathcal {H}_k(R)$$ , is a hypergraph with vertex set Z( R, k), and for distinct elements $$x_1,x_2,\ldots ,x_k$$ in Z( R, k), the set $$\{x_1,x_2, \ldots , x_k\}$$ is an edge of $$\mathcal {H}_k(R)$$ if and only if $$\prod \limits _{i=1}^kx_i =0$$ and the product of any $$(k-1)$$ elements of $$\{x_1,x_2,\ldots ,x_k\}$$ is nonzero. In this paper, we characterize all finite commutative non-local rings R with identity whose $$\mathcal {H}_3(R)$$ has crosscap one. [ABSTRACT FROM AUTHOR]

Published
2017
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43. Basic Concepts and Results

Author

van Maldeghem, Hendrik, Amann, H., editor, Bourguignon, J.-P., editor, Grove, K., editor, Lions, P.-L., editor, Araki, H., editor, Ball, J., editor, Brezzi, F., editor, Chang, K. C., editor, Hitchin, N., editor, Hofer, H., editor, Knörrer, H., editor, Masuda, K., editor, Zagier, D., editor, and van Maldeghem, Hendrik

Published
1998
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48. Classification of nonlocal rings with genus one 3-zero-divisor hypergraphs.

Author

Selvakumar, K. and Ramanathan, V.

Subjects
*COMMUTATIVE rings, *HYPERGRAPHS, *TOROIDAL harmonics, *PLANAR graphs, *DIVISOR theory
Abstract

LetRbe a commutative ring with identity and letZ(R,k) be the set of allk-zero-divisors inRandk>2 an integer. Thek-zero-divisor hypergraph ofR, denoted byℋk(R), is a hypergraph with vertex setZ(R,k), and for distinct elementsinZ(R,k), the setis an edge ofℋk(R) if and only ifand the product of any (k−1) elements ofis nonzero. In this paper, we characterize all finite commutative nonlocal ringsRwith identity whoseℋ3(R) has genus one. [ABSTRACT FROM AUTHOR]

Published
2017
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49. On the Metric Dimension of Imprimitive Distance-Regular Graphs.

Author

Bailey, Robert

Subjects
*METRIC spaces, *REGULAR graphs, *BIPARTITE graphs, *GEOMETRIC vertices, *ETHNOMATHEMATICS
Abstract

A resolving set for a graph $${\Gamma}$$ is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of $${\Gamma}$$ is the smallest size of a resolving set for $${\Gamma}$$ . Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distance-regular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs. We also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible. [ABSTRACT FROM AUTHOR]

Published
2016
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50. Connection and separation in hypergraphs

Author

Mohammad Bahmanian and Mateja Sajna

Subjects
hypergraph, incidence graph, walk, trail, path, cycle, connected hypergraph, cut edge, cut vertex, separating vertex, block, Mathematics, QA1-939
Abstract

In this paper we study various fundamental connectivity properties of hypergraphs from a graph-theoretic perspective, with the emphasis on cut edges, cut vertices, and blocks. We prove a number of new results involving these concepts. In particular, we describe the exact relationship between the block decomposition of a hypergraph and the block decomposition of its incidence graph.

Published
2015
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