1,675 research outputs found
On bipartite zero-divisor graphs
AbstractA (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We also prove that a zero-divisor graph is bipartite if and only if it contains no triangles. In addition, we give all corresponding zero-divisor semigroups of a class of complete bipartite graphs with a horn and determine which complete r-partite graphs with a horn have a corresponding semigroup for r≥3
Analisis Automorfisma Graf Pembagi-nol dari Ring Komutatif dengan Elemen Satuan
Zero-divisor graphs of a commutative ring with identity has 3 specific simple forms, namely star zero-divisor graph, complete zero-divisor graph and complete bipartite zero-divisor graph. Graph automorphism is one of the interesting concepts in graph theory. Automorphism of graph G is an isomorphism from graph G to itself. In other words, an automorphism of a graph G is a permutation φ of the set points V(G) which has the property that (x,y) in E(G) if and only if (φ(x),φ(y)) in E(G), i.e. φ preserves adjacency.This study aims to analyze the form of zero-divisor graph automorphisms of a commutative ring with identity formed. The method used in this study was taking sampel of each zero-divisor graph to represent each graph. Thus, pattern and shape of automorphism of each graph can be determined. Based on the results of this study, a star zero-divisor graph with pattern K_1,(p-1), where p is prime, has (p-1)! automorphisms, a complete zero-divisor graph with pattern K_(p-1), where p is prime, has (p-1)! automorphisms, and a complete bipartite zero-divisor graph with pattern K_(p-1),(q-1), where p is prime, has (p-1)!(q-1)! automorphisms, when p not equals to q and 2((p-1)!(q-1)!) automorphisms when p=q
Dimers and cluster integrable systems
We show that the dimer model on a bipartite graph on a torus gives rise to a
quantum integrable system of special type - a cluster integrable system. The
phase space of the classical system contains, as an open dense subset, the
moduli space of line bundles with connections on the graph. The sum of
Hamiltonians is essentially the partition function of the dimer model. Any
graph on a torus gives rise to a bipartite graph on the torus. We show that the
phase space of the latter has a Lagrangian subvariety. We identify it with the
space parametrizing resistor networks on the original graph.We construct
several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci.
EN
Pure simplicial complexes and well-covered graphs
A graph is called well-covered if all maximal independent sets of
vertices have the same cardinality. A simplicial complex is called
pure if all of its facets have the same cardinality. Let be the
class of graphs with some disjoint maximal cliques covering all vertices. In
this paper, we prove that for any simplicial complex or any graph, there is a
corresponding graph in class with the same well-coveredness
property. Then some necessary and sufficient conditions are presented to
recognize fast when a graph in the class is well-covered or not. To do
this characterization, we use an algebraic interpretation according to
zero-divisor elements of the edge rings of graphs.Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:1009.524
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