共查询到20条相似文献,搜索用时 15 毫秒
1.
Laurence R Matthews 《Journal of Combinatorial Theory, Series B》1979,27(3):260-273
A matroidal family is defined to be a collection of graphs such that, for any given graph G, the subgraphs of G isomorphic to a graph in satisfy the matroid circuit-axioms. Here matroidal families closed under homeomorphism are considered. A theorem of Simöes-Pereira shows that when only finite connected graphs are allowed as members of , two matroids arise: the cycle matroid and bicircular matroid. Here this theorem is generalized in two directions: the graphs are allowed to be infinite, and they are allowed to be disconnected. In the first case four structures result and in the second case two infinite families of matroids are obtained. The main theorem concerns the structures resulting when both restrictions are relaxed simultaneously. 相似文献
2.
Manfred Walter 《Discrete Mathematics》1982,41(3):309-315
A matroidal family of graphs is a set ≠Ø of connected finite graphs such that for every finite graph G the edge sets of those subgraphs of G which are isomorphic to some element of are the circuits of a matroid on the edge set of G. In [9], Schmidt shows that, for n?0, ?2n<r?1, n, , the set is a graph with β(G)=nα(G)+r and α(G )>, and is minimal with this property (with respect to the relation ?))} is a matroidal family of graphs. He also describes a method to construct new matroidal families of graphs by means of so-called partly closed sets. In this paper, an extension of this construction is given. By means of s-partly closed subsets of , s?r, we are able to give sufficient and necessary conditions for a subset of to yield a matroidal family of graphs when joined with the set of all graphs which satisfy: If , then H?G. In particular, it is shown that is not a matroidal family of graphs for r?2. Furthermore, for n?0, 1?2n<r, n, , the set of bipartite elements of can be used to construct new matroidal families of graphs if and only if s?min(n+r, 1). 相似文献
3.
For finite graphs F and G, let NF(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. If and are families of graphs, it is natural to ask then whether or not the quantities NF(G), F∈, are linearly independent when G is restricted to . For example, if = {K1, K2} (where Kn denotes the complete graph on n vertices) and is the family of all (finite) trees, then of course NK1(T) ? NK2(T) = 1 for all T∈. Slightly less trivially, if = {Sn: n = 1, 2, 3,…} (where Sn denotes the star on n edges) and again is the family of all trees, then Σn=1∞(?1)n+1NSn(T)=1 for all T∈. It is proved that such a linear dependence can never occur if is finite, no F∈ has an isolated point, and contains all trees. This result has important applications in recent work of L. Lovász and one of the authors (Graham and Lovász, to appear). 相似文献
4.
E.J Farrell 《Journal of Combinatorial Theory, Series B》1979,26(1):111-122
Let be a family of connected graphs. With each element α ∈ , we can associate a weight wα. Let G be a graph. An F-cover of G is a spanning subgraph of G in which every component belongs to . With every F-cover we can associate a monomial π(C) = Παwα, where the product is taken over all components of the cover. The F-polynomial of G is Σπ(C), where the sum is taken over all F-covers in G. We obtain general results for the complete graph and complete bipartite graphs, and we show that many of the well-known graph polynomials are special cases of more general F-polynomials. 相似文献
5.
Carsten Thomassen 《Journal of Combinatorial Theory, Series B》1982,33(2):137-160
Some basic results on duality of infinite graphs are established and it is proven that a block has a dual graph if and only if it is planar and any two vertices are separated by a finite edge cut. Also, the graphs having predual graphs are characterized completely and it is shown that if is a dual and predual graph of G, then G and can be represented as geometric dual graphs. The uniqueness of dual graphs is investigated, in particular, Whitney's 2-isomorphism theorem is extended to infinite graphs. Finally, infinite minimal cuts in dual graphs are studied and the characterization (in terms of planarity and separation properties) of the graphs having dual graphs satisfying conditions on the infinite cuts, as well, is included. 相似文献
6.
U. Cattaneo 《Journal of Functional Analysis》1980,35(2):143-152
The possibility of endowing an Abelian topological group G with the structure of a topological vector space when a subgroup F of G and the quotient group are topological vector groups is investigated. It is shown that, if F is a real Fréchet group and a complete metrizable real vector group, then G is a complete metrizable real vector group. This result is of particular interest if is finite dimensional or if F is one dimensional and a separable Hilbert group. 相似文献
7.
Seiya Negami 《Discrete Mathematics》1983,44(2):161-180
A topological generalization of the uniqueness of duals of 3-connected planar graphs will be obtained. A graph G is uniquely embeddable in a surface F if for any two embeddings , , there are an autohomeomorphism h:F → F and an automorphism σ:G → G such that . A graph G is faithfully embedabble in a surface F if there is an embedding such that for any automorphism σ:G → G, there is an autohomeomorphism h:F → F with . Our main theorems state that any 6-connected toroidal graph is uniquelly embeddable in a torus and that any 6-connected toroidal graph with precisely three exceptions is faithfully embeddable in a torus. The proofs are based on a classification of 6-regular torus graphs. 相似文献
8.
Let G be a finite abelian group. We investigate those graphs admitting G as a sharply 1-transitive automorphism group and all of whose eigenvalues are rational. The study is made via the rational algebra (G) of rational matrices with rational eigenvalues commuting with the regular matrix representation of G. In comparing the spectra obtainable for graphs in (G) for various G's, we relate subschemes of a related association scheme, subalgebras of (G), and the lattice of subgroups of G. One conclusion is that if the order of G is fifth-power-free, any graph with rational eigenvalues admitting G has a cospectral mate admitting the abelian group of the same order with prime-order elementary divisors. 相似文献
9.
In this note we demonstrate the existence of E0L forms F and G which are n-similar, i.e. n(F) = n(G) but n+1(F)≠n+1(G) for n ∈ {2, 3}. This partially solves an open problem from [3]. 相似文献
10.
P Frankl 《Journal of Combinatorial Theory, Series A》1977,22(2):249-251
The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2n. Then there is a family , || = m, such that F ∈ , G ? X, | G | > | F | implies G ∈ and minimizes the number of pairs (F1, F2), F1, F2 ∈ F1 ∩ F2 = ? over all families consisting of m subsets of X. 相似文献
11.
Properties of the graph of the polytope of all n × n nonnegative doubly stochastic matrices are studied. If is a face of which is not a k-dimensional rectangular parallelotope for k ≥ 2, then G() is Hamilton connected. Prime factor decompositions of the graphs of faces of relative to Cartesian product are investigated. In particular, if is a face of , then the number of prime graphs in any prime factor decomposition of G() equals the number of connected components of the neighborhood of any vertex of G(). Distance properties of the graphs of faces of are obtained. Faces of for which G() is a clique of are investigated. 相似文献
12.
Thomas Andreae 《Journal of Graph Theory》1978,2(2):149-153
A matroidal family is a nonempty set ? of connected finite graphs such that for every arbitrary finite graph G the edge sets of the subgraphs of G which are isomorphic to an element of ? form a matroid on the edge set of G. In the present paper the question whether there are any matroidal families besides the four previously described by Simões-Pereira is answered affirmatively. It is shown that for every natural number n ? 2 there is a matroidal family that contains the complete graph with n vertices. For n = 4 this settles Simões-Pereira's conjecture that there is a matroidal family containing all wheels. 相似文献
13.
Let be a family of subsets of S and let G be a graph with vertex set such that: (xA, xB) is an edge iff . The family is called a set representation of the graph G.It is proved that the problem of finding minimum k such that G can be represented by a family of sets of cardinality at most k is NP-complete. Moreover, it is NP-complete to decide whether a graph can be represented by a family of distinct 3-element sets.The set representations of random graphs are also considered. 相似文献
14.
Paul Milnes 《Journal of Mathematical Analysis and Applications》1978,65(1):32-43
Recently Lau [15] generalized a result of Yeadon [25]. In the present paper we generalize Yeadon's result in another direction recasting it as a theorem of ergodic type. We call the notion of ergodicity required left mean-ergodicity and show how it relates to the mean-ergodicity of Nagel [21]. Connections with the existence of invariant means on spaces of continuous functions on semitopological semigroups S are made, connections concerning, among other things, a fixed point theorem of Mitchell [20] and Schwartz's property P of W1-algebras [22]. For example, if (S) is a certain subspace of C(S) (which was considered by Mitchell and is of almost periodic type, i.e., the right translates of a member of (S) satisfy a compactness condition), then the assumption that (S) has a left invariant mean is equivalent to the assumption that every representation of S of a certain kind by operators on a linear topological space X is left mean-ergodic. An analog involving the existence of a (left and right) invariant mean on (S) is given, and we show our methods restrict in the Banach space setting to give short direct proofs of some results in [4], results involving the existence of an invariant mean on the weakly almost periodic functions on S or on the almost periodic functions on S. An ergodic theorem of Lloyd [16] is generalized, and a number of examples are presented. 相似文献
15.
Surinder K. Sehgal 《Journal of Number Theory》1974,6(2):124-127
16.
Michael Doob 《Journal of Combinatorial Theory, Series B》1974,17(3):205-217
A graph is magic if the edges are labeled with distinct nonnegative real numbers such that the sum of the labels incident to each vertex is the same. Given a graph finite G, an Abelian group , and an element r(v) ∈ for every v ∈ V(G), necessary and sufficient conditions are given for the existence of edge labels from such that the sum of the labels incident to v is r(v). When there do exist labels, all possible labels are determined. The matroid structure of the labels is investigated when is an integral domain, and a dimensional structure results. Characterizations of several classes of graphs are given, namely, zero magic, semi-magic, and trivial magic graphs. 相似文献
17.
Let λ(F) be the least eigenvalue of a finite graph F. The least limiting eigenvalue λ(G) of a connected infinite graph G is defined by λ(G)=infF{λ(F)}, where F runs over all finite induced subgraphs of G. In [4] and [5] it is proved that λ(G)⩾−2 if and only if G is a generalized line graph. In this paper all connected infinite graphs (thus all generalized line graphs) with λ(G)>−2 are characterized. 相似文献
18.
S.Bhaskara Rao 《Discrete Mathematics》1977,17(3):225-233
Let G be a self-complementary graph (s.c.) and π its degree sequence. Then G has a 2-factor if and only if π - 2 is graphic. This is achieved by obtaining a structure theorem regarding s.c. graphs without a 2-factor. Another interesting corollary of the structure theorem is that if G is a s.c. graph of order p?8 with minimum degree at least , then G has a 2-factor and the result is the best possible. 相似文献
19.
《Topology and its Applications》1986,22(2):109-122
We write 2x for the hyperspace of all non-empty compact sets in a complete metric linear space X topologized by the Hausdorff metric. Using the notation (X) = {A ϵ 2X: A is finite}, lf2 = {x} = (xi) ϵ l2: xi = 0 for almost all i}, and lσ2 = {x = (x i) ϵ l2:σ∞i=1 (ixi)2 < ∞}, we have the following theorem:A family ⊂(X) is homeomorphic to lf2 if is σ-fd-compact, the closure of in 2x is not locally compact and if whenever A, B ∈ , λ ∈ [0, 1] and C ⊂ λA + (1 - λ)B with card C⩽ max{card A, card B} then C ϵ .Moreover, for any Gδ-AR-set of with ⊃ we have (, )≅(l2, lƒ2).Similar conditions for hyperspaces to be homeomorphic to lσ2 are also established. 相似文献
20.
Lszl Surnyi 《Discrete Mathematics》1980,30(3):277-287
Let β(G) be the maximal β such that for any edge xy of G there is an independent β-set that contains no neighbours of x and y. Then and G is linecritical iff β(G) = α(G)?1. We determine the minimal connected graphs for any given β(G) or for any given β(G) and α(G). We study the case when β(G)??2 and give upper bounds for the minimal valencies. We generalize some results on linecritical graphs of [1] and [4]. 相似文献