共查询到20条相似文献,搜索用时 576 毫秒
1.
Stanislav Jendrol’ 《Discrete Mathematics》2006,306(24):3321-3326
The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which receive distinct colours. For a connected cubic plane graph G we prove that
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Vladimir R. Rosenfeld 《Discrete Applied Mathematics》2010,158(5):551-558
A stable (or independent) set in a graph is a set of pairwise nonadjacent vertices thereof. The stability numberα(G) is the maximum size of stable sets in a graph G. The independence polynomial of G is
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Walks and the spectral radius of graphs 总被引:1,自引:0,他引:1
Vladimir Nikiforov 《Linear algebra and its applications》2006,418(1):257-268
Given a graph G, write μ(G) for the largest eigenvalue of its adjacency matrix, ω(G) for its clique number, and wk(G) for the number of its k-walks. We prove that the inequalities
5.
Vladimir Nikiforov 《Linear algebra and its applications》2007,422(1):284-290
Suppose a graph G have n vertices, m edges, and t triangles. Letting λn(G) be the largest eigenvalue of the Laplacian of G and μn(G) be the smallest eigenvalue of its adjacency matrix, we prove that
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The higher Randi? index Rt(G) of a simple graph G is defined as
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John L. Goldwasser 《Discrete Mathematics》2008,308(12):2589-2593
The eternal domination number of a graph is the number of guards needed at vertices of the graph to defend the graph against any sequence of attacks at vertices. We consider the model in which at most one guard can move per attack and a guard can move across at most one edge to defend an attack. We prove that there are graphs G for which , where γ∞(G) is the eternal domination number of G and α(G) is the independence number of G. This matches the upper bound proved by Klostermeyer and MacGillivray. 相似文献
8.
For a connected graph G and any two vertices u and v in G, let D(u,v) denote the length of a longest u-v path in G. A hamiltonian coloring of a connected graph G of order n is an assignment c of colors (positive integers) to the vertices of G such that |c(u)−c(v)|+D(u,v)≥n−1 for every two distinct vertices u and v in G. The value of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number of G is taken over all hamiltonian colorings c of G. In this paper we discuss the hamiltonian chromatic number of graphs G with . As examples, we determine the hamiltonian chromatic number for a class of caterpillars, and double stars. 相似文献
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Dongmei Zhu 《Linear algebra and its applications》2010,432(11):2764-2772
In this paper, we obtain the following upper bound for the largest Laplacian graph eigenvalue λ(G):
11.
A stable set in a graph G is a set of mutually non-adjacent vertices, α(G) is the size of a maximum stable set of G, and is the intersection of all its maximum stable sets. It is known that if G is a connected graph of order n≥2 with 2α(G)>n, then , [V.E. Levit, E. Mandrescu, Combinatorial properties of the family of maximum stable sets of a graph, Discrete Applied Mathematics 117 (2002) 149-161; E. Boros, M.C. Golumbic, V.E. Levit, On the number of vertices belonging to all maximum stable sets of a graph, Discrete Applied Mathematics 124 (2002) 17-25]. When we restrict ourselves to the class of trees, we add some structural properties to this statement. Our main finding is the theorem claiming that if T is a tree of order n≥2, with 2α(T)>n, then at least two pendant vertices an even distance apart belong to . 相似文献
12.
Vladimir Nikiforov 《Linear algebra and its applications》2006,414(1):347-360
Let G be a graph with n vertices and m edges and let μ(G) = μ1(G) ? ? ? μn(G) be the eigenvalues of its adjacency matrix. Set s(G)=∑u∈V(G)∣d(u)-2m/n∣. We prove that
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Bc(G) denotes the cyclic bandwidth of graph G. In this paper, we obtain the maximum cyclic bandwidth of graphs of order p with adding an edge as follows:
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Yoshiharu Kohayakawa Vojtěch Rödl Mathias Schacht Papa Sissokho 《Journal of Combinatorial Theory, Series A》2007,114(4):631-657
The generalized Turán number ex(G,H) of two graphs G and H is the maximum number of edges in a subgraph of G not containing H. When G is the complete graph Km on m vertices, the value of ex(Km,H) is , where o(1)→0 as m→∞, by the Erd?s-Stone-Simonovits theorem.In this paper we give an analogous result for triangle-free graphs H and pseudo-random graphs G. Our concept of pseudo-randomness is inspired by the jumbled graphs introduced by Thomason [A. Thomason, Pseudorandom graphs, in: Random Graphs '85, Poznań, 1985, North-Holland, Amsterdam, 1987, pp. 307-331. MR 89d:05158]. A graph G is (q,β)-bi-jumbled if
16.
Let G be a graph. Then the hamiltonian index h(G) of G is the smallest number of iterations of line graph operator that yield a hamiltonian graph. In this paper we show that for every 2-connected simple graph G that is not isomorphic to the graph obtained from a dipole with three parallel edges by replacing every edge by a path of length l≥3. We also show that for any two 2-connected nonhamiltonian graphs G and with at least 74 vertices. The upper bounds are all sharp. 相似文献
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Shengbiao Hu 《Discrete Mathematics》2007,307(2):280-284
Let G be a simple graph. Let λ1(G) and μ1(G) denote the largest eigenvalue of the adjacency matrix and the Laplacian matrix of G, respectively. Let Δ denote the largest vertex degree. If G has just one cycle, then
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Inverse degree and edge-connectivity 总被引:2,自引:0,他引:2
Let G be a connected graph with vertex set V(G), order n=|V(G)|, minimum degree δ and edge-connectivity λ. Define the inverse degree of G as , where d(v) denotes the degree of the vertex v. We show that if
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An independent set of a graph G is a set of pairwise non-adjacent vertices. Let α(G) denote the cardinality of a maximum independent set and fs(G) for 0≤s≤α(G) denote the number of independent sets of s vertices. The independence polynomial defined first by Gutman and Harary has been the focus of considerable research recently. Wingard bounded the coefficients fs(T) for trees T with n vertices: for s≥2. We generalize this result to bounds for a very large class of graphs, maximal k-degenerate graphs, a class which includes all k-trees. Additionally, we characterize all instances where our bounds are achieved, and determine exactly the independence polynomials of several classes of k-tree related graphs. Our main theorems generalize several related results known before. 相似文献