共查询到20条相似文献,搜索用时 31 毫秒
1.
The well known correspondence between even cycles of an undirected graph and polynomials in a binomial ideal associated to a graph is extended to odd cycles and polynomials in another binomial ideal. Other binomial ideals associated to an undirected graph are also introduced. The results about them with topics on monomial ideals are used in order to show decision procedures for bipartite graphs, minimal vertex covers, cliques, edge covers and matchings with algebraic tools. All such procedures are implemented in Maple 9.5. 相似文献
2.
In this paper we consider three classes of chain hexagonal cacti and study their matching and independence related properties. Explicit recurrences are derived for their matching and independence polynomials, and explicit formulae are presented for the number of matchings and independents sets of certain types. Bivariate generating functions for the number of matchings and independent sets of certain types are also computed and then used to deduce the expected size of matchings and independent sets in chains of given length. It is shown that the extremal chain hexagonal cacti with respect to the number of matchings and of independent sets belong to one of the considered types. Possible directions of further research are discussed. 相似文献
3.
Generalized matrix tree theorem for mixed graphs 总被引:11,自引:0,他引:11
Ravindra B. Bapat Jerrold W. Grossman Devadatta M. Kulkarni 《Linear and Multilinear Algebra》1999,46(4):299-312
4.
K. Steffens 《Combinatorica》1985,5(4):359-365
The Edmonds—Gallai decomposition theorem for matchings of finite or locally finite graphs is generalized to matchings of the
kernel of an arbitrary graph. 相似文献
5.
In graph theory, the related problems of deciding when a set of vertices or a set of edges constitutes a maximum matching or a minimum covering have been extensively studied. In this paper we generalize these ideas by defining total matchings and total coverings, and show that these sets, whose elements in general consist of both vertices and edges, provide a way to unify these concepts. Parameters denoting the maximum and the minimum cardinality of these sets are introduced and upper and lower bounds depending only on the order of the graph are obtained for the number of elements in arbitrary total matchings and total coverings. Precise values of all the parameters are found for several general classes of graphs, and these are used to establish the sharpness of most of the bounds. In addition, variations of some well known equalities due to Gallai relating covering and matching numbers are obtained. 相似文献
6.
In this paper, we find lower bounds for the maximum and minimum numbers of cliques in maximal sets of pairwise disjoint cliques in a graph. By complementation, these yield lower bounds for the maximum and minimum numbers of independent sets in maximal sets of pairwise disjoint maximal independent sets of vertices in a graph. In the latter context, we show by examples that one of our bounds is best possible. 相似文献
7.
The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices and neither perfect matchings nor almost-perfect matchings. In this paper, we prove general results regarding the matching preclusion number and the conditional matching preclusion number as well as the classification of their respective optimal sets for regular graphs. We then use these general results to study the problems for Cayley graphs generated by 2-trees and the hyper Petersen networks. 相似文献
8.
A strong clique in a graph is a clique intersecting every maximal independent set. We study the computational complexity of six algorithmic decision problems related to strong cliques in graphs and almost completely determine their complexity in the classes of chordal graphs, weakly chordal graphs, line graphs and their complements, and graphs of maximum degree at most three. Our results rely on connections with matchings and relate to several graph properties studied in the literature, including well-covered graphs, localizable graphs, and general partition graphs. 相似文献
9.
10.
An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p≥3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete. 相似文献
11.
We derive closed formulas for the numbers of independent sets of size at most 4 and matchings of size at most 3 in graphs without small cycles that depend only on the degree sequence and the products of the degrees of adjacent vertices.
As a related problem we describe an algorithm that determines a tree of given degree sequence that maximizes the sum of the products of the degrees of adjacent vertices. 相似文献
12.
The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject.In the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a Riemann-Hilbert analysis.In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees. 相似文献
13.
We adapt the cycle space of a finite or locally finite
graph to graphs with vertices of infinite degree, using as
cycles the homeomorphic
images of the unit circle S1 in the
graph together with its ends. We characterize the spanning trees
whose fundamental cycles generate this cycle space, and prove
infinite analogues to the standard characterizations of finite
cycle spaces in terms of edge-decomposition into single cycles
and orthogonality to cuts.To the memory of C. St. J. A.
Nash-Williams 相似文献
14.
We adapt the cycle space of a finite graph to locally
finite infinite graphs, using as infinite cycles the
homeomorphic images of the unit circle S1 in the
graph compactified by its ends. We prove that this cycle space
consists of precisely the sets of edges that meet every finite
cut evenly, and that the spanning trees whose fundamental cycles
generate this cycle space are precisely the end-faithful
spanning trees. We also generalize Eulers theorem by showing
that a locally finite connected graph with ends contains a
closed topological curve traversing every edge exactly once if
and only if its entire edge set lies in this cycle space.To the memory of C. St. J. A.
Nash-Williams 相似文献
15.
Let G be a finite undirected graph without loops and multiple edges. A graph is called “unterringfrei”, if there is no induced subgraph being a cycle of length?4. This note shows that the chromatic polynomial of such a graph does only have positive integers as its roots. Conversely all polynomials with only positive integral roots are the chromatic polynomials of such graphs under the slight restriction that M(G;α)=0 implies that M(G;β)=0 for α>β?0. 相似文献
16.
Došli? and Måløy (2010) [2] obtained the extremal 6-cactus chains with respect to the number of matchings and of independent sets. Motivated by the prior paper, in this paper we give recurrences for matching polynomials of ortho-chains and meta-chains, and show that they are the h-cactus chains with the most matchings. 相似文献
17.
Daniel Horsley 《Annals of Combinatorics》2012,16(3):571-589
We prove that a complete bipartite graph can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, the length of each cycle is at most the size of the smallest part, and the longest cycle is at most three times as long as the second longest. We then use this result to obtain results on incomplete even cycle systems with a hole and on decompositions of complete multipartite graphs into cycles of uniform even length. 相似文献
18.
R.H. Jeurissen 《Discrete Mathematics》1975,13(3):251-260
We link some well-known theorems and prove some new ones, each on one or more of the items of the title, and together illustrating their close relationship. The main tools are well-known similar conditions on maximum stable sets and maximum matchings, by which we prove theorems on the existence of odd cycles, including a generalization of Konig's equality between the matching and covering numbers of a bipartite graph. We deal with the question “How nearly bipartite is a graph?” and conjecture an inequality involving the matching and covering numbers and the number of disjoint odd cycles. 相似文献
19.
20.
We prove positivity results about linearization and connection coefficients for Bessel polynomials. The proof is based on
a recursion formula and explicit formulas for the coefficients in special cases. The result implies that the distribution
of a finite convex combination of independent Student t-variables with arbitrary odd degrees of freedom has a density which
is a finite convex combination of certain Student t-densities with odd degrees of freedom. 相似文献