Rubber bands,convex embeddings and graph connectivity

We give various characterizations ofk-vertex connected graphs by geometric, algebraic, and physical properties. As an example, a graphG isk-connected if and only if, specifying anyk vertices ofG, the vertices ofG can be represented by points of ^k–1 so that nok are on a hyper-plane and each vertex is in the convex hull of its neighbors, except for thek specified vertices. The proof of this theorem appeals to physics. The embedding is found by letting the edges of the graph behave like ideal springs and letting its vertices settle in equilibrium.As an algorithmic application of our results we give probabilistic (Monte-Carlo and Las Vegas) algorithms for computing the connectivity of a graph. Our algorithms are faster than the best known (deterministic) connectivity algorithms for allkn, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.     

Generalized characteristic polynomials of graph bundles

In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic polynomial of the graph. Since the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, consequently, the Bartholdi zeta function of a graph bundle can be computed by using our computational formulae.     

The tree number of a graph with a given girth

A family of subtrees of a graphG whose edge sets form a partition of the edge set ofG is called atree decomposition ofG. The minimum number of trees in a tree decomposition ofG is called thetree number ofG and is denoted by(G). It is known that ifG is connected then(G) |G|/2. In this paper we show that ifG is connected and has girthg 5 then(G) |G|/g + 1. Surprisingly, the case wheng = 4 seems to be more difficult. We conjecture that in this case(G) |G|/4 + 1 and show a wide class of graphs that satisfy it. Also, some special graphs like complete bipartite graphs andn-dimensional cubes, for which we determine their tree numbers, satisfy it. In the general case we prove the weaker inequality(G) (|G| – 1)/3 + 1.     

On a geometric property of perfect graphs

LetG be a graph,VP(G) its vertex packing polytope and letA(G) be obtained by reflectingVP(G) in all Cartersian coordinates. Denoting byA*(G) the set obtained similarly from the fractional vertex packing polytope, we prove that the segment connecting any two non-antipodal vertices ofA(G) is contained in the surface ofA(G) and thatG is perfect if and only ifA*(G) has a similar property.     

On the connectedness of the complement of the zero-divisor graph of a poset

Abstract

In this paper, connectedness is completely characterized for the complements of the zero-divisor graphs of partially ordered sets. These results are applied to annihilating ideal graphs and intersection graphs of submodules, generalizing some of the work that has recently appeared in the literature.     

Monotone path systems in simple regions

A monotone path system (MPS) is a finite set of pairwise disjoint paths (polygonal areas) in thexy-plane such that every horizontal line intersects each of the paths in at most one point. A MPS naturally determines a pairing of its top points with its bottom points. We consider a simple polygon in thexy-plane wich bounds the simple polygonal (closed) regionD. LetT andB be two finite, disjoint, equicardinal sets of points ofD. We give a good characterization for the existence of a MPS inD which pairsT withB, and a good algorithm for finding such a MPS, and we solve the problem of finding all MPSs inD which pairT withB. We also give sufficient conditions for any such pairing to be the same.The first author's research is supported by the Natural Sciences and Engineering Research Council of Canada     

On the full automorphism group of a graph

While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ _p wrZ _p , almost all Cayley graphs ofG have automorphism group isomorphic toG.     

A generalized characteristic polynomial of a graph having a semifree action

For an abelian group Γ, a formula to compute the characteristic polynomial of a Γ-graph has been obtained by Lee and Kim [Characteristic polynomials of graphs having a semi-free action, Linear algebra Appl. 307 (2005) 35-46]. As a continuation of this work, we give a computational formula for generalized characteristic polynomial of a Γ-graph when Γ is a finite group. Moreover, after showing that the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, we compute the reciprocals of the Bartholdi zeta functions of wheels and complete bipartite graphs as an application of our formula.     

The hypermetric cone is polyhedral

The hypermetric coneH _n is the cone in the spaceR ^n(n–1)/2 of all vectorsd=(d _ij)_1i<jn satisfying the hypermetric inequalities: –_1ijn z _j z _j d _ij 0 for all integer vectorsz inZ ⁿ with –_1in z _i =1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres andL-polytopes in lattices). As an application, we show that the hypermetric coneH _n is polyhedral.     

Cones of alternating and cut submodular set functions

We describe facets of the cones of alternating set functions and cut submodular set functions generated by directed and undirected graphs and by uniform even hypergraphs. This answers a question asked by L. Lovász at the Bonn Mathematical Programming Conference in 1982. We show that there is a network flow algorithm for minimizing a hypergraph cut set function.     

Zeta functions of line, middle, total graphs of a graph and their coverings

We consider the (Ihara) zeta functions of line graphs, middle graphs and total graphs of a regular graph and their (regular or irregular) covering graphs. Let L(G), M(G) and T(G) denote the line, middle and total graph of G, respectively. We show that the line, middle and total graph of a (regular and irregular, respectively) covering of a graph G is a (regular and irregular, respectively) covering of L(G), M(G) and T(G), respectively. For a regular graph G, we express the zeta functions of the line, middle and total graph of any (regular or irregular) covering of G in terms of the characteristic polynomial of the covering. Also, the complexities of the line, middle and total graph of any (regular or irregular) covering of G are computed. Furthermore, we discuss the L-functions of the line, middle and total graph of a regular graph G.     

Pairs of edge-disjoint Hamiltonian circuits

Aequationes mathematicae -     

Geradenanordnungen Mit Maximaler Anzahl Vierseitiger Flächen

Ohne Zusammenfassung

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Search for properties of the missing Moore graph

In the degree-diameter problem, the only extremal graph the existence of which is still in doubt is the Moore graph of order 3250, degree 57 and diameter 2. It has been known that such a graph cannot be vertex-transitive. Also, certain restrictions on the structure of the automorphism group of such a graph have been known in the case when the order of the group is even. In this paper we further investigate symmetries and structural properties of the missing Moore (57, 2)-graph(s) with the help of a combination of spectral, group-theoretic, combinatorial, and computational methods. One of the consequences is that the order of the automorphism group of such a graph is at most 375.     

Puzzles and polytope isomorphisms

We show that an isomorphism between the graphs of two simple polytopes of arbitrary dimension can always be extended to an isomorphism between the polytopes themselves. It has been convenient to study the dual situation, involving what we like to call the puzzle of a simplicial polytope.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

The spectral radii of a graph and its line graph

The spectral radius of a (directed) graph is the largest eigenvalue of adjacency matrix of the (directed) graph. We give the relation on the characteristic polynomials of a directed graph and its line graph, and obtain sharp bounds on the spectral radius of directed graphs. We also give the relation on the spectral radii of a graph and its line graph. As a consequence, the spectral radius of a connected graph does not exceed that of its line graph except that the graph is a path.     

On affine motions and universal rigidity of tensegrity frameworks

Recently, Alfakih and Ye (2013) [4] proved that if an r  -dimensional bar framework (G,p)$(G,p)$ on n?r+2$n?r+2$ nodes in general position in R^r${\mathbb{R}}^r$ admits a positive semidefinite stress matrix with rank n−r−1$n-r-1$, then (G,p)$(G,p)$ is universally rigid. In this paper, we generalize this result in two directions. First, we extend this result to tensegrity frameworks. Second, we replace the general position assumption by the weaker assumption that in configuration p, each point and its neighbors in G   affinely span R^r${\mathbb{R}}^r$.     

Low diameter graph decompositions

Adecomposition of a graphG=(V,E) is a partition of the vertex set into subsets (calledblocks). Thediameter of a decomposition is the leastd such that any two vertices belonging to the same connected component of a block are at distance d. In this paper we prove (nearly best possible) statements, of the form: Anyn-vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [4] it was shown that every graph has a decomposition into at mosts(n) blocks of diameter at mosts(n) for . Using a technique of Awerbuch [3] and Awerbuch and Peleg [5], we improve this result by showing that every graph has a decomposition of diameterO (logn) intoO(logn) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in timeO(log² n). The construction can be parameterized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible, for two families of graphs: the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner's lemma for the first class and Tucker's lemma for the second.A preliminary version of this paper appeared as Decomposing Graphs into Regions of Small Diameter in Proc. 2nd ACM-SIAM Symposium on Discrete Algorithms (1991) 321-330.This work was supported in part by NSF grant DMS87-03541 and by a grant from the Israel Academy of Science.This work was supported in part by NSF grant DMS87-03541 and CCR89-11388.     

The least eigenvalue of a graph with cut vertices

In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut vertices, and then obtain a lower bound for the least eigenvalue of a connected graph in terms of the number of cut vertices. During the discussion we also get some results for the spectral radius of a connected bipartite graph and its upper bound.