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1.
2.
Daqing Yang 《Discrete Mathematics》2009,309(13):4614-4623
Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,
1.
if and then or (fraternity);
2.
if and then (transitivity).
In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col2(G)≤O(1(G)0(G)2), where k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that k(G) bounds the distance (k+1)-coloring number colk+1(G) with a function f(k(G)). On the other hand, k(G)≤(col2k+1(G))2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.  相似文献   

3.
Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γk(G), the connected k-domination number ; the k-independent domination number and the k-irredundance number irk(G). The authors prove that if an irk-set X is a k-independent set of G, then , and that for k?2, if irk(G)=1, if irk(G) is odd, and if irk(G) is even, which generalize some known results.  相似文献   

4.
Let rk(n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p,
  相似文献   

5.
The higher Randi? index Rt(G) of a simple graph G is defined as
  相似文献   

6.
Xuding Zhu 《Discrete Mathematics》2009,309(18):5562-5568
Given a graph G and a positive integer p, χp(G) is the minimum number of colours needed to colour the vertices of G so that for any ip, any subgraph H of G of tree-depth i gets at least i colours. This paper proves an upper bound for χp(G) in terms of the k-colouring number of G for k=2p−2. Conversely, for each integer k, we also prove an upper bound for in terms of χk+2(G). As a consequence, for a class K of graphs, the following two statements are equivalent:
(a)
For every positive integer p, χp(G) is bounded by a constant for all GK.
(b)
For every positive integer k, is bounded by a constant for all GK.
It was proved by Nešet?il and Ossona de Mendez that (a) is equivalent to the following:
(c)
For every positive integer q, q(G) (the greatest reduced average density of G with rank q) is bounded by a constant for all GK.
Therefore (b) and (c) are also equivalent. We shall give a direct proof of this equivalence, by introducing q−(1/2)(G) and by showing that there is a function Fk such that . This gives an alternate proof of the equivalence of (a) and (c).  相似文献   

7.
Let T(G) be the number of spanning trees in graph G. In this note, we explore the asymptotics of T(G) when G is a circulant graph with given jumps.The circulant graph is the 2k-regular graph with n vertices labeled 0,1,2,…,n−1, where node i has the 2k neighbors i±s1,i±s2,…,i±sk where all the operations are . We give a closed formula for the asymptotic limit as a function of s1,s2,…,sk. We then extend this by permitting some of the jumps to be linear functions of n, i.e., letting si, di and ei be arbitrary integers, and examining
  相似文献   

8.
We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c?1 of deletion:
(1)
, , , and .
(2)
For all k?2, and .
(3)
For all k?2, .
(4)
.
(5)
For all k?2, .
For many of these results, even the c=1 case was not previously known.Similar to the definition of reconstruction numbers vrn(G) [F. Harary, M. Plantholt, The graph reconstruction number, J. Graph Theory 9 (1985) 451-454] and ern(G) (see [J. Lauri, R. Scapellato Topics in Graph Automorphism and Reconstruction, London Mathematical Society, Cambridge University Press, Cambridge, 2003, p. 120]), we introduce two new graph parameters, vrn(G) and ern(G), and give an example of a family {Gn}n?4 of graphs on n vertices for which vrn(Gn)<vrn(Gn). For every k?2 and n?1, we show that there exists a collection of k graphs on (2k-1+1)n+k vertices with 2n 1-vertex-preimages, i.e., one has families of graph collections whose number of 1-vertex-preimages is huge relative to the size of the graphs involved.  相似文献   

9.
Let n(G) denote the number of vertices of a graph G and let α(G) be the independence number of G, the maximum number of pairwise nonadjacent vertices of G. The Hall ratio of a graph G is defined by
  相似文献   

10.
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:
  相似文献   

11.
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
  相似文献   

12.
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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13.
14.
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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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≤sα(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   

17.
A.R. Rao 《Discrete Mathematics》2006,306(14):1595-1600
For a digraph G, let R(G) (respectively, R(k)(G)) be the number of ordered pairs (u,v) of vertices of G such that uv and v is reachable from u (respectively, reachable from u by a path of length ?k). In this paper, we study the range Sn of R(G) and the range of R(k)(G) as G varies over all possible digraphs on n vertices. We give a sufficient condition and a necessary condition for an integer to belong to Sn. These determine the set Sn for all n?208. We also determine for k?4 and show that whenever n?k+(k+1)0.57+2, for arbitrary k.  相似文献   

18.
Let G be a locally compact group and let B(G) be the dual space of C∗(G), the group C∗ algebra of G. The Fourier algebra A(G) is the closed ideal of B(G) generated by elements with compact support. The Fourier algebras have a natural operator space structure as preduals of von Neumann algebras. Given a completely bounded algebra homomorphism we show that it can be described, in terms of a piecewise affine map with Y in the coset ring of H, as follows
  相似文献   

19.
Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G) = D(G) + A(G). In [5], Cvetkovi? et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (q2) and the index (λ1) of graph G (see also Aouchiche and Hansen [1]):
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20.
In the paper we examine Pexiderized ?-homogeneity equation almost everywhere. Assume that G and H are groups with zero, (X,G) and (Y,H) are a G- and an H-space, respectively. We prove, under some assumption on (Y,H), that if functions and satisfy Pexiderized ?-homogeneity equation
F1(αx)=?(α)F2(x)  相似文献   

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