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The generalized -connectivity of a graph was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized -edge-connectivity which is defined as and , where denotes the maximum number of pairwise edge-disjoint trees in such that for . In this paper we prove that for any two connected graphs and we have , where is the Cartesian product of and . Moreover, the bound is sharp. We also obtain the...
We study the generalized -connectivity as introduced by Hager in 1985, as well as the more recently introduced generalized -edge-connectivity . We determine the exact value of and for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case .
Let be a graph. A vertex subversion strategy of , say , is a set of vertices in whose closed neighborhood is removed from . The survival-subgraph is denoted by . The Neighbor-Integrity of , , is defined to be , where is any vertex subversion strategy of , and is the maximum order of the components of . In this paper we give some results connecting the neighbor-integrity and binary graph operations.
The paper studies graphs in which each pair of vertices has exactly two common neighbours. It disproves a conjectury by P. Hliněný concerning these graphs.
The generalized k-connectivity κk(G) of a graph G, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized k-edge-connectivity λk(G). In this paper, graphs of order n such that [...] for even k are characterized.
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