Graph Onn Research Articles

On integer matrices with integer eigenvalues and Laplacian integral graphs

A matrix A is said to be an integer matrix if all its entries are integers. In this article, we characterize the nonsingular integer matrices with integer eigenvalues in an expressible form of their corresponding inverse matrices. It is proved that a nonsingular integer matrix A has integer eigenvalues if and only if A−1 can be written as the sum of n rank-one matrices that meet certain requirements. A method for constructing integer matrices with integer eigenvalues using the Hadamard product is also provided. Let S represent an n-tuple of nonnegative integers. If there is an n×n integer matrix A whose spectrum (the collection of eigenvalues) is S, we say that S is realisable by an integer matrix. In Fallat et al. (2005) [7], the authors posed a conjecture that “there is no simple graph onn≥2vertices whose Laplacian spectrum is given by(0,1,…,n−1).” We provide a characterization of threshold graphs using the spectra of quotient matrices of its G-join graphs. As a consequence, we prove that given any n−1 positive integers λ2,…,λn such that λ2≤⋯≤λn, the n-tuple (0,λ2,…,λn) is realizable by the Laplacian matrix of a multidigraph. In particular, we show that (0,1,…,n−1) can be realizable by the Laplacian matrix of a multidigraph.

Extremal unicyclic graphs with respect to additively weighted Harary index

In this paper we define cycle-star graphCSk;n k to be a graph onn vertices consisting of the cycle of lengthk andn k leafs appended to the same vertex of the cycle. Also, we define cycle-path graphCPk;n k to be a graph onn vertices consisting of the cycle of lengthk and of path on n k vertices whose one end is linked to a vertex on a cycle. We establish that cycle- star graphCS3;n 3 is the only maximal graph with respect to additively weighted Harary index among all unicyclic graphs on n vertices, while cycle-path graph CP3;n 3 is the only minimal unicyclic graph (here n must be at least 5): The values of additively weighted Harary index for extremal unicyclic graphs are established, so these values are the upper and the lower bound for the value of additively weighted Harary index on the class of unicyclic graphs onn vertices. 2010 Mathematics Subject Classification: 05C35

Near-optimal bounded-degree spanning trees

Random costsC(i, j) are assigned to the arcs of a complete directed graph onn labeled vertices. Given the cost matrixC n =(C(i, j)), letT* k =T* k (C n ) be the spanning tree that has minimum cost among spanning trees with in-degree less than or equal tok. Since it is NP-hard to findT* k , we instead consider an efficient algorithm that finds a near-optimal spanning treeT . If the edge costs are independent, with a common exponential(I) distribution, then, asn → ∞, $$E(Cost(T_k^a {\text{)) = }}E(Cost(T_k^* {\text{)) + }}o\left( 1 \right).$$ Upper and lower bounds forE(Cost(T* k )) are also obtained fork≥2.

On-line coloringk-colorable graphs

We show that for anyk, there exists an on-line algorithm that will color anyk-colorable graph onn vertices withO(n1−1/k!) colors. This improves the previous best upper bound ofO(nlog(2k−3)n/log(2k−4)n) due to Lovasz, Saks, and Trotter. In the special casesk=3 andk=4 we obtain on-line algorithms that useO(n2/3log1/3n) andO(n5/6log1/6n) colors, respectively.

Approximating the independence number via theϑ-function

We describe an approximation algorithm for the independence number of a graph. If a graph onn vertices has an independence numbern/k + m for some fixed integerk ⩾ 3 and somem > 0, the algorithm finds, in random polynomial time, an independent set of size\(\tilde \Omega (m^{{3 \mathord{\left/ {\vphantom {3 {(k + 1)}}} \right. \kern-\nulldelimiterspace} {(k + 1)}}} )\), improving the best known previous algorithm of Boppana and Halldorsson that finds an independent set of size Ω(m1/(k−1)) in such a graph. The algorithm is based on semi-definite programming, some properties of the Lovaszϑ-function of a graph and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If theϑ-function of ann vertex graph is at leastMn1−2/k for some absolute constantM, we describe another, related, efficient algorithm that finds an independent set of sizek. Several examples show the limitations of the approach and the analysis together with some related arguments supply new results on the problem of estimating the largest possible ratio between theϑ-function and the independence number of a graph onn vertices. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

On the minimal number of edges in color-critical graphs

A graphG isk-critical if it has chromatic numberk, but every proper subgraph of it is (k−1)-colorable. This paper is devoted to investigating the following question: for givenk andn, what is the minimal number of edges in ak-critical graph onn vertices, with possibly some additional restrictions imposed? Our main result is that for everyk≥4 andn>k this number is at least $$\left( {\frac{{k - 1}}{2} + \frac{{k - 3}}{{2(k^2 - 2k - 1)}}} \right)n$$ , thus improving a result of Gallai from 1963. We discuss also the upper bounds on the minimal number of edges ink-critical graphs and provide some constructions of sparsek-critical graphs. A few applications of the results to Ramsey-type problems and problems about random graphs are described.

On-line coloring of perfect graphs

Lovasz, Saks, and Trotter showed that there exists an on-line algorithm which will color any on-linek-colorable graph onn vertices withO(nlog(2k−3) n/log(2k−4) n) colors. Vishwanathan showed that at least Ω(log k−1 n/k k ) colors are needed. While these remain the best known bounds, they give a distressingly weak approximation of the number of colors required. In this article we study the case of perfect graphs. We prove that there exists an on-line algorithm which will color any on-linek-colorable perfect graph onn vertices withn 10k/loglogn colors and that Vishwanathan's techniques can be slightly modified to show that his lower bound also holds for perfect graphs. This suggests that Vishwanathan's lower bound is far from tight in the general case.

The random-cluster model on the complete graph

The random-cluster model of Fortuin and Kasteleyn contains as special cases the percolation, Ising, and Potts models of statistical physics. When the underlying graph is the complete graph onn vertices, then the associated processes are called ‘mean-field’. In this study of the mean-field random-cluster model with parametersp=λ/n andq, we show that its properties for any value ofq∈(0, ∞) may be derived from those of an Erdős-Renyi random graph. In this way we calculate the critical pointλ c (q) of the model, and show that the associated phase transition is continuous if and only ifq≦2. Exact formulae are given forλ C (q), the density of the largest component, the density of edges of the model, and the ‘free energy’. This work generalizes earlier results valid for the Potts model, whereq is an integer satisfyingq≧2. Equivalent results are obtained for a ‘fixed edge-number’ random-cluster model. As a consequence of the results of this paper, one obtains large-deviation theorems for the number of components in the classical random-graph models (whereq=1).

Independent sets, cliques and hamiltonian graphs

LetG be ak-connected (k ≥ 2) graph onn vertices. LetS be an independent set ofG. S is called essential if there exist two distinct vertices inS which have a common neighbor inG. LetV r, be a clique which is a complete subgraph ofG. In this paper it is proven that if every essential independent setS ofk + 1 vertices satisfiesS ∩ V r ≠ ∅, thenG is hamiltonian, orG∖{u} is hamiltonian for someu ∈ V r, orG is one of three classes of exceptional graphs. Our theorem generalizes several well-known theorems.

Degree sums,k-factors and hamilton cycles in graphs

We prove (a generalization of) the following conjecture of R. Haggkvist: LetG be a 2-connected graph onn vertices where every pair of nonadjacent vertices has degree sum at leastn — k and assume thatG has ak-factor; thenG is hamiltonian. This result is a common generalization of well-known theorems of Ore and Jackson, respectively.

Recursive construction for 3-regular expanders

We present an algorithm which inn 3 (logn)3 time constructs a 3-regular expander graph onn vertices. In each step we substitute a pair of edges of the graph by a new pair of edges so that the total number of cycles of lengths=⌊clogn⌋ decreases (for some fixed absolute constantc). When we reach a local minimum in the number of cycles of lengths the graph is an expander.

Small cycle double covers of 4-connected planar graphs

Acycle double cover of a graph,G, is a collection of cycles,C, such that every edge ofG lies in precisely two cycles ofC. TheSmall Cycle Double Cover Conjecture, proposed by J. A. Bondy, asserts that every simple bridgeless graph onn vertices has a cycle double cover with at mostn−1 cycles, and is a strengthening of the well-knownCycle Double Cover Conjecture. In this paper, we prove Bondy's conjecture for 4-connected planar graphs.

Rigorous results on the thermodynamics of the dilute Hopfield model

We study the Hopfield model of an autoassociative memory on a random graph onN vertices where the probability of two vertices being joined by a link isp(N). Assuming thatp(N) goes to zero more slowly thanO(1/N), we prove the following results: (1) If the number of stored patternsm(N) is small enough such thatm(N)/Np(N) ↓0, asN↑∞, then the free energy of this model converges, upon proper rescaling, to that of the standard Curie-Weiss model, for almost all choices of the random graph and the random patterns. (2) If in additionm(N) < ln N/ln 2, we prove that there exists, forT< 1, a Gibbs measure associated to each original pattern, whereas for higher temperatures the Gibbs measure is unique. The basic technical result in the proofs is a uniform bound on the difference between the Hamiltonian on a random graph and its mean value.

The spectrum of maximal partial steiner triple systems

The determination of the possible numbers of triples in a maximal partial triple system of order λ begun by Severn is completed. The result rests on the characterization of the maximum number of edges in a triangle-free, nonbipartite, antieulerian graph onn vertices.

The cut cone III: On the role of triangle facets

The cut polytopeP n is the convex hull of the incidence vectors of the cuts (i.e. complete bipartite subgraphs) of the complete graph onn nodes. A well known class of facets ofP n arises from the triangle inequalities:x ij +x ik +x jk ≤ 2 andx ij -x ik -x jk ≤ 0 for 1 ≤i,j, k ≤n. Hence, the metric polytope Mn, defined as the solution set of the triangle inequalities, is a relaxation ofP n . We consider several properties of geometric type for Pn, in particular, concerning its position withinM n . Strengthening the known fact ([3]) thatP n has diameter 1, we show that any set ofk cuts,k ≤ log2 n, satisfying some additional assumption, determines a simplicial face ofM n and thus, also, ofP n . In particular, the collection of low dimension faces ofP n is contained in that ofM n . Among a large subclass of the facets ofP n , the triangle facets are the closest ones to the barycentrum of Pn and we conjecture that this result holds in general. The lattice generated by all even cuts (corresponding to bipartitions of the nodes into sets of even cardinality) is characterized and some additional questions on the links between general facets ofP n and its triangle facets are mentioned.

Cycles through specified vertices

Recently, various authors have obtained results about the existence of long cycles in graphs with a given minimum degreed. We extend these results to the case where only some of the vertices are known to have degree at leastd, and we want to find a cycle through as many of these vertices as possible. IfG is a graph onn vertices andW is a set ofw vertices of degree at leastd, we prove that there is a cycle through at least\(\left\lceil {\frac{w}{{\left\lceil {{n \mathord{\left/ {\vphantom {n d}} \right. \kern-\nulldelimiterspace} d}} \right\rceil - 1}}} \right\rceil \) vertices ofW. We also find the extremal graphs for this property.

Facets for the cut cone II: Clique-web inequalities

We study new classes of facets for the cut coneC n generated by the cuts of the complete graph onn vertices. This cone can also be interpreted as the cone of all semi-metrics onn points that are isometricallyl 1-embeddable and, in fact, the study of the facets of the cut polytope is in some sense equivalent to the study of the facets ofC n . These new facets belong to the class of clique-web inequalities which generalize the hypermetric and cycle inequalities as well as the bicycle odd wheel inequalities.

Facets for the cut cone I

We study facets of the cut coneC n , i.e., the cone of dimension 1/2n(n − 1) generated by the cuts of the complete graph onn vertices. Actually, the study of the facets of the cut cone is equivalent in some sense to the study of the facets of the cut polytope. We present several operations on facets and, in particular, a “lifting” procedure for constructing facets ofC n+1 from given facets of the lower dimensional coneC n . After reviewing hypermetric valid inequalities, we describe the new class of cycle inequalities and prove the facet property for several subclasses. The new class of parachute facets is developed and other known facets and valid inequalities are presented.

On Hajnal's triangle-free game

We investigate the triangle-free game proposed by Andras Hajnal. Starting with the empty graph onn points, two players alternatingly pick edges. The loser is the player who is forced to select an edge which completes a triangle. We determine the winner in a version of the game with the additional rule that the chosen edges must always give a connected subgraph ofK n . Some other versions are also investigated.

Cycles in weighted graphs

A weighted graph is one in which each edgee is assigned a nonnegative numberw(e), called the weight ofe. The weightw(G) of a weighted graphG is the sum of the weights of its edges. In this paper, we prove, as conjectured in [2], that every 2-edge-connected weighted graph onn vertices contains a cycle of weight at least 2w(G)/(n−1). Furthermore, we completely characterize the 2-edge-connected weighted graphs onn vertices that contain no cycle of weight more than 2w(G)/(n−1). This generalizes, to weighted graphs, a classical result of Erdős and Gallai [4].