Bipartite Divisor Graphs for Integer Subsets

Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph Γ(X) and the prime vertex graph Δ(X), for a set X of positive integers, we define the bipartite divisor graph B(X), and show that many of these connections flow naturally from properties of B(X). In particular we establish links between parameters of these three graphs, such as number and diameter of components, and we characterise bipartite graphs that can arise as B(X) for some X. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of B(X), for one of Γ(X) or Δ(X) to contain a complete subgraph of size 3 or 4.     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

On the Commuting Graph of Dihedral Group

Let Γ be a non-abelian group and Ω ? Γ. We define the commuting graph G = 𝒞(Γ, Ω) with vertex set Ω and two distinct elements of Ω are joined by an edge when they commute in Γ. In this article, among some properties of commuting graphs, we investigate distant properties as well as detour distant properties of commuting graph on D_2n. We also study the metric dimension of commuting graph on D_2n and compute its resolving polynomial.     

Extended F4-buildings and the Baby Monster

Let Τ be the Baby Monster graph which is the graph on the set of {3,4}-transpositions in the Baby Monster group B in which two such transpositions are adjacent if their product is a central involution in B. Then Τ is locally the commuting graph of central (root) involutions in ² E ₆(2). The graph Τ contains a family of cliques of size 120. With respect to the incidence relation defined via inclusion these cliques and the non-empty intersections of two or more of them form a geometry ℰ(B) with diagram for t=4 and the action of B on ℰ(B) is flag-transitive. We show that ℰ(B) contains subgeometries ℰ(² E ₆(2)) and ℰ(Fi ₂₂) with diagrams c.F ₄(2) and c.F ₄(1). The stabilizers in B of these subgeometries induce on them flag-transitive actions of ² E ₆(2):2 and Fi ₂₂:2, respectively. The geometries ℰ(B), ℰ(² E ₆(2)) and ℰ(Fi ₂₂) possess the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2 and (b) three elements of type 1 are pairwise incident to common elements of type 2 if and only if they are incident to a common element of type 5. The paper addresses the classification problem of c.F ₄(t)-geometries satisfying (a) and (b). We construct three further examples for t=2 with flag-transitive automorphism groups isomorphic to 3⋅²E₂:2, E₆(2):2 and 2²⁶ .F₄(2) and one for t=1 with flag-transitive automorphism group 3⋅Fi ₂₂:2. We also study the graph of an arbitrary (non-necessary flag-transitive) c.F ₄(t)-geometry satisfying (a) and (b) and obtain a complete list of possibilities for the isomorphism type of subgraph induced by the common neighbours of a pair of vertices at distance 2. Finally, we prove that ℰ(B) is the only c.F ₄(4)-geometry, satisfying (a) and (b). Oblatum 20-X-1999 & 2-I-2001?Published online: 5 March 2001     

Upper bounds for the diameter and height of graphs of convex polyhedra

Let Δ(d, n) be the maximum diameter of the graph of ad-dimensional polyhedronP withn-facets. It was conjectured by Hirsch in 1957 that Δ(d, n) depends linearly onn andd. However, all known upper bounds for Δ(d, n) were exponential ind. We prove a quasi-polynomial bound Δ(d, n)≤n ^{2 logd+3}. LetP be ad-dimensional polyhedron withn facets, let ϕ be a linear objective function which is bounded onP and letv be a vertex ofP. We prove that in the graph ofP there exists a monotone path leading fromv to a vertex with maximal ϕ-value whose length is at most . This research was supported in part by a BSF grant, by a GIF grant, and by ONR-N00014-91-C0026.     

Finding Scores in Tournaments

A tournamentT_nis an orientation of the complete graph onnvertices. We continue the algorithmic study initiated by10of recognizing various directed trees in tournaments. Hell and Rosenfeld studied the complexity of finding various oriented paths in tournaments by probing edge directions. Here, we investigate the complexity of finding a vertex of prescribed outdegree (or indegree) in the same model. We show that the complexity of finding a vertex of outdegreek( ≤ (n − 1)/2) inT_nis Θ(nk). This bound is in sharp contrast to the Θ(n) bound for selection in the case of transitive tournaments. We also establish tight bounds for finding vertices of prescribed degree from the adjacency matrix of general directed/undirected graphs. These bounds generalize the classical bound of11for finding a sink (a vertex of outdegree 0 and indegreen − 1) in a directed graph.     

On Some Subfactors of Integer Index Arising from Vertex Models

This paper is devoted to the study of subfactors arising out of commuting squares constructed out of the simplest possible vertex models. After setting up the necessary general background, we start with two classes of commuting squares and compute the principal graphs of the resulting subfactors, one of them being related to the group dual of a suitable (closed) subgroup ofU(N), and the other to the Cayley graph of a (non-closed) group, modulo scalars, generated byNelements ofU(N). In the last section, we “classify” the commuting squares in dimension 2, and identify the possible resulting principal graphs asA⁽¹⁾_(2n−1), 1n∞     

The diameter of paired-domination vertex critical graphs

In this paper we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998), 199–206). A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ _pr(G), is the minimum cardinality of a paired-dominating set of G. The graph G is paired-domination vertex critical if for every vertex v of G that is not adjacent to a vertex of degree one, γ _pr(G − v) < γ _pr(G). We characterize the connected graphs with minimum degree one that are paired-domination vertex critical and we obtain sharp bounds on their maximum diameter. We provide an example which shows that the maximum diameter of a paired-domination vertex critical graph is at least 3/2 (γ _pr(G) − 2). For γ _pr(G) ⩽ 8, we show that this lower bound is precisely the maximum diameter of a paired-domination vertex critical graph. The first author was supported in part by the South African National Research Foundation and the University of KwaZulu-Natal, the second author was supported by the Natural Sciences and Engineering Research Council of Canada.     

The Conditional Covering Problem on Unweighted Interval Graphs with Nonuniform Coverage Radius

Let G = (V, E) be an interval graph with n vertices and m edges. A positive integer R(x) is associated with every vertex x ? V{x\in V}. In the conditional covering problem, a vertex x ? V{x \in V} covers a vertex y ? V{y \in V} (x ≠ y) if d(x, y) ≤ R(x) where d(x, y) is the shortest distance between the vertices x and y. The conditional covering problem (CCP) finds a minimum cardinality vertex set C í V{C\subseteq V} so as to cover all the vertices of the graph and every vertex in C is also covered by another vertex of C. This problem is NP-complete for general graphs. In this paper, we propose an efficient algorithm to solve the CCP with nonuniform coverage radius in O(n ²) time, when G is an interval graph containing n vertices.     

On the total domination subdivision numbers in graphs

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_γt (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Karami, Khoeilar, Sheikholeslami and Khodkar, (Graphs and Combinatorics, 2009, 25, 727–733) proved that for any connected graph G of order n ≥ 3, sd_{γ t} (G) ≤ 2γ _t (G) − 1 and posed the following problem: Characterize the graphs that achieve the aforementioned upper bound. In this paper we first prove that sd_{γ t} (G) ≤ 2α′(G) for every connected graph G of order n ≥ 3 and δ(G) ≥ 2 where α′(G) is the maximum number of edges in a matching in G and then we characterize all connected graphs G with sd_{γ t} (G)=2γ _t (G)−1.     

Matching behaviour is asymptotically normal

Ak-matching in a graphG is a set ofk edges, no two of which have a vertex in common. The number of these inG is writtenp(G, k). Using an idea due to L. H. Harper, we establish a condition under which these numbers are approximately normally distributed. We show that our condition is satisfied ifn=|V(G)| is large compared to the maximum degree Δ of a vertex inG(i.e. Δ=o(n)) orG is a large complete graph. One corollary of these results is that the number of points fixed by a randomly chosen involution in the symmetric groupS is asymptotically normally distributed.     

Commuting graphs of groups and related numerical parameters

Given a finite group G, we denote by Δ(G) the commuting graph of G which is defined as follows: the vertex set is G and two distinct vertices x and y are joined by an edge if and only if xy = yx. Clearly, Δ(G) is always connected for any group G. We denote by κ(G) the number of spanning trees of Δ(G). In the present paper, among other results, we first obtain the value κ(G) for some specific groups G, such as Frobenius groups, Dihedral groups, AC-groups, etc. Next, we characterize the alternating group A₅, in the class of nonsolvable groups through its tree-number κ(A₅). Finally, we classify the finite groups for which the power graph and the commuting graph coincide.     

Counterexamples to a conjecture of mader about cycles through specified vertices inn-edge-connected graphs

For each oddn≥3, we constructn-edge-connected graphsG with the following property: There are two verticesu andv inG such that for every cycleC inG passign throughu andv the graphG-E(C) is not (n-2)-edge-connected. HereE(C) denotes the set of edges ofC, and a cycle is allowed to pass through a vertex more than once.     

On second order degree of graphs

Given a vertex v of a graph G the second order degree of v denoted as d 2(v) is defined as the number of vertices at distance 2 from v.In this paper we address the following question:What are the sufficient conditions for a graph to have a vertex v such that d2(v) ≥ d(v),where d(v) denotes the degree of v? Among other results,every graph of minimum degree exactly 2,except four graphs,is shown to have a vertex of second order degree as large as its own degree.Moreover,every K-4-free graph or every maximal planar graph is shown to have a vertex v such that d2(v) ≥ d(v).Other sufficient conditions on graphs for guaranteeing this property are also proved.     

The commuting graphs of some subsets in the quaternion algebra over the ring of integers modulo <Emphasis Type="Italic">N</Emphasis>

Let R be an arbitrary ring, S be a subset of R, and Z(S) = {s ∈ S | sx = xs for every x ∈ S}. The commuting graph of S, denoted by Γ(S), is the graph with vertex set S \ Z(S) such that two different vertices x and y are adjacent if and only if xy = yx. In this paper, let I _n , N _n be the sets of all idempotents, nilpotent elements in the quaternion algebra ℤ _n [i, j, k], respectively. We completely determine Γ(I _n ) and Γ(N _n ). Moreover, it is proved that for n ≥ 2, Γ(I _n ) is connected if and only if n has at least two odd prime factors, while Γ(N _n ) is connected if and only if n ∈ 2, 2², p, 2p for all odd primes p.     

Set Partition Complexes

The Hom complexes were introduced by Lovász to study topological obstructions to graph colorings. The vertices of Hom(G,K _n ) are the n-colorings of the graph G, and a graph coloring is a partition of the vertex set into independent sets. Replacing the independence condition with any hereditary condition defines a set partition complex. We show how coloring questions arising from, for example, Ramsey theory can be formulated with set partition complexes. It was conjectured by Babson and Kozlov, and proved by Čukić and Kozlov, that Hom(G,K _n ) is (n−d−2)-connected, where d is the maximal degree of a vertex of G. We generalize this to set partition complexes.     

An interlacing theorem for matrices whose graph is a given tree

Let A and B be (n×n)-matrices. For an index set S ⊂ {1, …, n}, denote by A(S) the principal submatrix that lies in the rows and columns indexed by S. Denote by S′ the complement of S and define η(A, B) = det A(S) det B(S′), where the summation is over all subsets of {1, …, n} and, by convention, det A(∅) = det B(∅) = 1. C. R. Johnson conjectured that if A and B are Hermitian and A is positive semidefinite, then the polynomial η(λA,-B) has only real roots. G. Rublein and R. B. Bapat proved that this is true for n ⩽ 3. Bapat also proved this result for any n with the condition that both A and B are tridiagonal. In this paper, we generalize some little-known results concerning the characteristic polynomials and adjacency matrices of trees to matrices whose graph is a given tree and prove the conjecture for any n under the additional assumption that both A and B are matrices whose graph is a tree. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 245–254, 2004.     

Expanders obtained from affine transformations

A bipartite graphG=(U, V, E) is an (n, k, δ, α) expander if |U|=|V|=n, |E|≦kn, and for anyX⊆U with |X|≦αn, |Γ _G (X)|≧(1+δ(1−|X|/n)) |X|, whereΓ _G (X) is the set of nodes inV connected to nodes inX with edges inE. We show, using relatively elementary analysis in linear algebra, that the problem of estimating the coefficientδ of a bipartite graph is reduced to that of estimating the second largest eigenvalue of a matrix related to the graph. In particular, we consider the case where the bipartite graphs are defined from affine transformations, and obtain some general results on estimating the eigenvalues of the matrix by using the discrete Fourier transform. These results are then used to estimate the expanding coefficients of bipartite graphs obtained from two-dimensional affine transformations and those obtained from one-dimensional ones.     

Super Connectivity and Super Edge Connectivity of the Mycielskian of a Graph

Mycielski introduced a new graph transformation μ(G) for graph G, which is called the Mycielskian of G. A graph G is super connected or simply super-κ (resp. super edge connected or super-λ), if every minimum vertex cut (resp. minimum edge cut) isolates a vertex of G. In this paper, we show that for a connected graph G with |V(G)| ≥ 2, μ(G) is super-κ if and only if δ(G) < 2κ(G), and μ(G) is super-λ if and only if G\ncong K₂{G\ncong K_2}.