Weakly central-vertex complete graphs with applications to commutative rings

The zero-divisor graph of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy=0. In this paper, a decomposition theorem is provided to describe weakly central-vertex complete graphs of radius 1. This characterization is then applied to the class of zero-divisor graphs of commutative rings. For finite commutative rings whose zero-divisor graphs are not isomorphic to that of Z₄[X]/(X²), it is shown that weak central-vertex completeness is equivalent to the annihilator condition. Furthermore, a schema for describing zero-divisor graphs of radius 1 is provided.     

A lower estimate for the achromatic number of irreducible graphs

The achromatic number of a graph is the largest number of independent sets its vertex set can be split into in such a way that the union of any two of these sets is not independent. A graph is irreducible if no two vertices have the same neighborhood. The achromatic number of an irreducible graph with n vertices is shown to be $\text{≥(}\text{1}\text{2}\text{?0(1))log}\text{n?}\text{log log}\text{n,}$ while an example of Erdös shows that it need not be log n/log 2+2 for any n. The proof uses an indiscernibility argument.     

Graphs with three mutually pseudo-similar vertices

Vertices u and v of a graph X are pseudo-similar if X ? u ? X ? v but no automorphism of X maps u to v. We describe a group-theoretic method for constructing graphs with a set of three mutually pseudo-similar vertices. The method is used to construct several examples of such graphs. An algorithm for extending, in a natural way, certain graphs with three mutually pseudo-similar vertices to a graph in which the three vertices are similar is given. The algorithm suggests that no simple characterization of graphs with a set of three mutually pseudo-similar vertices can exist.     

Half-Transitive Graphs of Prime-Cube Order

We call an undirected graph X half-transitive if the automorphism group Aut X of X acts transitively on the vertex set and edge set but not on the set of ordered pairs of adjacent vertices of X. In this paper we determine all half-transitive graphs of order p ³ and degree 4, where p is an odd prime; namely, we prove that all such graphs are Cayley graphs on the non-Abelian group of order p ³ and exponent p ², and up to isomorphism there are exactly (p – 1)/2 such graphs. As a byproduct, this proves the uniqueness of Holt's half-transitive graph with 27 vertices.     

Distance-balanced graphs: Symmetry conditions

A graph X is said to be distance-balanced if for any edge uv of X, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. A graph X is said to be strongly distance-balanced if for any edge uv of X and any integer k, the number of vertices at distance k from u and at distance k+1 from v is equal to the number of vertices at distance k+1 from u and at distance k from v. Exploring the connection between symmetry properties of graphs and the metric property of being (strongly) distance-balanced is the main theme of this article. That a vertex-transitive graph is necessarily strongly distance-balanced and thus also distance-balanced is an easy observation. With only a slight relaxation of the transitivity condition, the situation changes drastically: there are infinite families of semisymmetric graphs (that is, graphs which are edge-transitive, but not vertex-transitive) which are distance-balanced, but there are also infinite families of semisymmetric graphs which are not distance-balanced. Results on the distance-balanced property in product graphs prove helpful in obtaining these constructions. Finally, a complete classification of strongly distance-balanced graphs is given for the following infinite families of generalized Petersen graphs: GP(n,2), GP(5k+1,k), GP(3k±3,k), and GP(2k+2,k).     

On the generation of bicliques of a graph

An independent set of a graph is a subset of pairwise non-adjacent vertices. A complete bipartite set B is a subset of vertices admitting a bipartition B=X∪Y, such that both X and Y are independent sets, and all vertices of X are adjacent to those of Y. If both X,Y≠∅, then B is called proper. A biclique is a maximal proper complete bipartite set of a graph. When the requirement that X and Y are independent sets of G is dropped, we have a non-induced biclique. We show that it is NP-complete to test whether a subset of the vertices of a graph is part of a biclique. We propose an algorithm that generates all non-induced bicliques of a graph. In addition, we propose specialized efficient algorithms for generating the bicliques of special classes of graphs.     

Theory of path length distributions I

The Path Length Distribution (PLD) of a (p, q) graph is defined to be the array (X₀, X₁, X₂, …, X_p-1), where X₀ is the number of (unordered) pairs of vertices which have no path connecting them and X_l, 1 ≦ l ≦ p-1, is the number of pairs of vertices which are connected by a path of length l (see [1, 2]). The topic of this paper is the occurence of non-isomorphic graphs having the same path length distribution. For trees, a constructive procedure is given, showing that for any positive integer N there exist N non-isomorphic trees of diameter four which have the same PLD. Also considered are PLD-maximal graphs — those graphs with p vertices such that all pairs of vertices are connected by a path of length l for 2 ≦ l ≦ p-1. In addition to providing more examples of non-isomorphic graphs having the same PLD, PLD-maximal graphs are of intrinsic interest. For PLD-maximal graphs, we give sufficient degree and edge conditions and a necessary edge condition.     

On the X-join decomposition for undirected graphs

In his paper [17], Sabidussi defined the X-join of a family of graphs. Cowan, James, Stanton gave in [6] and O(n⁴) algorithm that decomposes a graph, when possible, into the X-join of the family of its subgraphs. We give here another approach using an equivalence relation on the edge set of the graph. We prove that if G and its complement are connected then there exists an unique class of edges that covers all the vertices of G. This theorem yields immediately an O(n³) decomposition algorithm.     

Geometric Achromatic and Pseudoachromatic Indices

The pseudoachromatic index of a graph is the maximum number of colors that can be assigned to its edges, such that each pair of different colors is incident to a common vertex. If for each vertex its incident edges have different color, then this maximum is known as achromatic index. Both indices have been widely studied. A geometric graph is a graph drawn in the plane such that its vertices are points in general position, and its edges are straight-line segments. In this paper we extend the notion of pseudoachromatic and achromatic indices for geometric graphs, and present results for complete geometric graphs. In particular, we show that for n points in convex position the achromatic index and the pseudoachromatic index of the complete geometric graph are \(\lfloor \frac{n^2+n}{4} \rfloor \).     

Achromatic number of fragmentable graphs

A complete coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at least one edge. The achromatic number ψ(G) is the greatest number of colors in such a coloring. We say a class of graphs is fragmentable if for any positive ε, there is a constant C such that any graph in the class can be broken into pieces of size at most C by removing a proportion at most ε of the vertices. Examples include planar graphs and grids of fixed dimension. Determining the achromatic number of a graph is NP‐complete in general, even for trees, and the achromatic number is known precisely for only very restricted classes of graphs. We extend these classes very considerably, by giving, for graphs in any class which is fragmentable, triangle‐free, and of bounded degree, a necessary and sufficient condition for a sufficiently large graph to have a complete coloring with a given number of colors. For the same classes, this gives a tight lower bound for the achromatic number of sufficiently large graphs, and shows that the achromatic number can be determined in polynomial time. As examples, we give exact values of the achromatic number for several graph families. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:94–114, 2010     

Graphs with 1-Hamiltonian-connected cubes

The cube G³ of a connected graph G is that graph having the same vertex set as G and in which two distinct vertices are adjacent if and only if their distance in G is at most three. A Hamiltonian-connected graph has the property that every two distinct vertices are joined by a Hamiltonian path. A graph G is 1-Hamiltonian-connected if, for every vertex w of G, the graphs G and G?w are Hamiltonian-connected. A characterization of graphs whose cubes are 1-Hamiltonian-connected is presented.     

On 3-Steiner simplicial orderings

Let G be a connected graph and S a nonempty set of vertices of G. A Steiner tree for S is a connected subgraph of G containing S that has a minimum number of edges. The Steiner interval for S is the collection of all vertices in G that belong to some Steiner tree for S. Let k≥2 be an integer. A set X of vertices of G is k-Steiner convex if it contains the Steiner interval of every set of k vertices in X. A vertex x∈X is an extreme vertex of X if X?{x} is also k-Steiner convex. We call such vertices k-Steiner simplicial vertices. We characterize vertices that are 3-Steiner simplicial and give characterizations of two classes of graphs, namely the class of graphs for which every ordering produced by Lexicographic Breadth First Search is a 3-Steiner simplicial ordering and the class for which every ordering of every induced subgraph produced by Maximum Cardinality Search is a 3-Steiner simplicial ordering.     

Algebraic multigrid methods for Laplacians of graphs

Classical algebraic multigrid theory relies on the fact that the system matrix is positive definite. We extend this theory to cover the positive semidefinite case as well, by formulating semiconvergence results for these singular systems. For the class of irreducible diagonal dominant singular M-matrices we show that the requirements of the developed theory hold and that the coarse level systems are still of the same class, if the C/F-splitting is good enough. An important example for matrices that are irreducible diagonal dominant M-matrices are Laplacians of graphs. Recent shape optimizing methods for graph partitioning require to solve singular linear systems involving these Laplacians. We present convergence results as well as experimental results for numerous graphs arising from finite element discretizations with up to 10⁶ vertices.     

On the diameter of the intersection graph of a finite simple group

Let G be a finite group. The intersection graph Δ_G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected.     

Regular graphs with high edge degree

For any vertex x of a graph G let Δ(x) denote the set of vertices adjacent to x. We seek to describe the connected graphs G which are regular of valence n and in which for all adjacent vertices x and y |Δ(x) ∩ Δ(y)| = n ? 1 ? s. It is known that the complete graphs are the graphs for which s = 0. For any s, any complete many-partite graph, each part containing s + 1 vertices, is such a graph. We show that these are the only such graphs for which the valence exceeds 2s² ? s + 1. The graphs satisfying these conditions for s = 1 or 2 are characterized (up to the class of trivalent triangle-free graphs.)     

Super-connected edge transitive graphs

A graph G is said to be super-connected if any minimum cut of G isolates a vertex. In a previous work due to the second author of this note, super-connected graphs which are both vertex transitive and edge transitive are characterized. In this note, we generalize the characterization to edge transitive graphs which are not necessarily vertex transitive, showing that the only irreducible edge transitive graphs which are not super-connected are the cycles C_n(n?6) and the line graph of the 3-cube, where irreducible means the graph has no vertices with the same neighbor set. Furthermore, we give some sufficient conditions for reducible edge transitive graphs to be super-connected.     

On a conjecture of Brouwer involving the connectivity of strongly regular graphs

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components.We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called Δ-spaces are counterexamples to Brouwer?s Conjecture. Using J.I. Hall?s characterization of finite reduced copolar spaces, we find that the triangular graphs T(m), the symplectic graphs Sp(2r,q) over the field Fq (for any q prime power), and the strongly regular graphs constructed from the hyperbolic quadrics O⁺(2r,2) and from the elliptic quadrics O⁻(2r,2) over the field F₂, respectively, are counterexamples to Brouwer?s Conjecture. For each of these graphs, we determine precisely the minimum number of vertices whose removal disconnects the graph into non-singleton components. While we are not aware of an analogue of Hall?s characterization theorem for Δ-spaces, we show that complements of the point graphs of certain finite generalized quadrangles are point graphs of Δ-spaces and thus, yield other counterexamples to Brouwer?s Conjecture.We prove that Brouwer?s Conjecture is true for many families of strongly regular graphs including the conference graphs, the generalized quadrangles GQ(q,q) graphs, the lattice graphs, the Latin square graphs, the strongly regular graphs with smallest eigenvalue −2 (except the triangular graphs) and the primitive strongly regular graphs with at most 30 vertices except for few cases.We leave as an open problem determining the best general lower bound for the minimum size of a disconnecting set of vertices of a strongly regular graph, whose removal disconnects the graph into non-singleton components.     

Diameters of random circulant graphs

The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters increase linearly in the number of nodes. In the present study we consider an intermediate class of examples: Cayley graphs of cyclic groups, also known as circulant graphs or multi-loop networks. We show that the diameter of a random circulant 2k-regular graph with n vertices scales as n ^1/k , and establish a limit theorem for the distribution of their diameters. We obtain analogous results for the distribution of the average distance and higher moments.     

A trust region method for solving the decentralized static output feedback design problem

The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents anO(n ²) time sequential algorithm and anO(n ²/p+logn) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, wherep andn represent respectively the number of processors and the number of vertices of the circular-arc graph.     

On the spectral radius of bicyclic graphs with n vertices and diameter d

Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. In this paper we determine the graph with the largest spectral radius among all bicyclic graphs with n vertices and diameter d. As an application, we give first three graphs among all bicyclic graphs on n vertices, ordered according to their spectral radii in decreasing order.