共查询到20条相似文献,搜索用时 546 毫秒
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Two graphs are said to be -cospectral (respectively, -cospectral) if they have the same (respectively, signless) Laplacian spectra, and a graph is said to be (respectively, ) if there does not exist other non-isomorphic graph such that and are -cospectral (respectively, -cospectral). Let be the degree sequence of a graph with vertices. In this paper, we prove that except for two exceptions (respectively, the graphs with ), if is -cospectral (respectively, -cospectral) with a connected graph and , then has the same degree sequence as . A spider graph is a unicyclic graph obtained by attaching some paths to a common vertex of the cycle. As an application of our result, we show that every spider graph and its complement graph are both , which extends the corresponding results of Haemers et al. (2008), Liu et al. (2011), Zhang et al. (2009) and Yu et al. (2014). 相似文献
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Susan A. van Aardt Christoph Brause Alewyn P. Burger Marietjie Frick Arnfried Kemnitz Ingo Schiermeyer 《Discrete Mathematics》2017,340(11):2673-2677
An edge-coloured graph is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph denoted by , is the smallest number of colours that are needed in order to make properly connected. Our main result is the following: Let be a connected graph of order and . If , then except when and where and 相似文献
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We say a graph is -colorable with of ’s and of ’s if may be partitioned into independent sets and sets whose induced graphs have maximum degree at most . The maximum average degree, , of a graph is the maximum average degree over all subgraphs of . In this note, for nonnegative integers , we show that if , then is -colorable. 相似文献
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Bart Litjens 《Discrete Mathematics》2018,341(6):1740-1748
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Ping Sun 《Discrete Mathematics》2012,312(24):3649-3655
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Let denote the chromatic polynomial of a graph on vertices. The ‘shameful conjecture’ due to Bartels and Welsh states that, Let denote the expected number of colors used in a uniformly random proper -coloring of . The above inequality can be interpreted as saying that , where is the empty graph on nodes. This conjecture was proved by F.M. Dong, who in fact showed that, for all . There are examples showing that this inequality is not true for all . In this paper, we show that the above inequality holds for all , where is the largest degree of . It is also shown that the above inequality holds true for all when is a claw-free graph. 相似文献
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Let be a -connected graph of order . In [1], Bondy (1980) considered a degree sum condition for a graph to have a Hamiltonian cycle, say, to be covered by one cycle. He proved that if , then has a Hamiltonian cycle. On the other hand, concerning a degree sum condition for a graph to be covered by two cycles, Enomoto et al. (1995) [4] proved that if and , then can be covered by two cycles. By these results, we conjecture that if , then can be covered by two cycles. In this paper, we prove the case of this conjecture. In fact, we prove a stronger result; if is 2-connected with , then can be covered by two cycles, or belongs to an exceptional class. 相似文献
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This paper deals with the Cayley graph , where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that is the product of the left translation group and a dihedral group of order . The proof uses several properties of the subgraph of induced by the set . In particular, is a -regular graph whose automorphism group is
has as many as maximal cliques of size , and its subgraph whose vertices are those in these cliques is a 3-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of of order with regular Cayley maps on is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non--balanced regular Cayley map on . 相似文献
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《Applied Mathematics Letters》2006,19(4):345-350
Let and be two hamiltonian paths of . We say that and are independent if , and for . We say a set of hamiltonian paths of between two distinct vertices are mutually independent if any two distinct paths in the set are independent. We use to denote the number of vertices and use to denote the number of edges in graph . Moreover, we use to denote the number of edges in the complement of . Suppose that is a graph with and . We prove that there are at least mutually independent hamiltonian paths between any pair of distinct vertices of except and . Assume that is a graph with the degree sum of any two non-adjacent vertices being at least . Let and be any two distinct vertices of . We prove that there are mutually independent hamiltonian paths between and if and there are mutually independent hamiltonian paths between and if otherwise. 相似文献