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71.
David Ellison 《Discrete Mathematics》2019,342(3):917-925
In the game of cops and robbers on graphs, the cops and the robber are allowed to pass their turn if they are located on a looped vertex. This paper explores the effect of loops on the cop number and the capture time. We provide examples of graphs where the cop number almost doubles when the loops are removed, graphs where the cop number decreases when the loops are removed, graphs where the capture time is quadratic in the number of vertices and copwin graphs where the cop needs to move away from the robber in optimal play. 相似文献
72.
A graph is -colorable if it admits a vertex partition into a graph with maximum degree at most and a graph with maximum degree at most . We show that every -free planar graph is -colorable. We also show that deciding whether a -free planar graph is -colorable is NP-complete. 相似文献
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The Laplacian spectrum of a graph consists of the eigenvalues (together with multiplicities) of the Laplacian matrix. In this article we determine, among the graphs consisting of disjoint unions of paths and cycles, those ones which are determined by the Laplacian spectrum. For the graphs, which are not determined by the Laplacian spectrum, we give the corresponding cospectral non-isomorphic graphs. 相似文献
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Ariane Carrance 《Random Structures and Algorithms》2019,55(3):615-648
We present here random distributions on (D + 1)‐edge‐colored, bipartite graphs with a fixed number of vertices 2p. These graphs encode D‐dimensional orientable colored complexes. We investigate the behavior of those graphs as p→∞. The techniques involved in this study also yield a Central Limit Theorem for the genus of a uniform map of order p, as p→∞. 相似文献
80.
《Discrete Mathematics》2019,342(10):2770-2782
“Which graphs are determined by their spectrum (DS for short)?” is a fundamental question in spectral graph theory. It is generally very hard to show a given graph to be DS and few results about DS graphs are known in literature. In this paper, we consider the above problem in the context of the generalized -spectrum. A graph is said to be determined by the generalized -spectrum (DGQS for short) if, for any graph , and have the same -spectrum and so do their complements imply that is isomorphic to . We give a simple arithmetic condition for a graph being DGQS. More precisely, let be a graph with adjacency matrix and degree diagonal matrix . Let be the signless Laplacian matrix of , and ( is the all-ones vector) be the -walk matrix. We show that if (which is always an integer) is odd and square-free, then is DGQS. 相似文献