Authors: Z. Jin, F. Li, Y. Sun Title: On two generalized connectivities of graphs Source: Discussiones Mathematicae Graph Theory Received 17.06.2016, Revised 02.11.2016, Accepted 07.11.2016, doi: 10.7151/dmgt.1987 | |
Abstract: The concept of generalized k-connectivity κ_{k}(G), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. The pendant tree-connectivity τ_{k}(G) was also introduced by Hager in 1985, which is a specialization of generalized k-connectivity but a generalization of the classical connectivity. Another generalized connectivity of a graph G, named k-connectivity κ'_{k}(G), introduced by Chartrand et al. in 1984, is a generalization of the cut-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of κ'_{k}(G) and τ_{k}(G) by showing that for a connected graph G of order n, if κ'_{k}(G)≠ n-k+1 where k≥ 3, then 1≤ κ'_{k}(G)-τ_{k}(G)≤ n-k; otherwise, 1≤ κ'_{k}(G)-τ_{k}(G)≤ n-k+1. Moreover, all of these bounds are sharp. We get a sharp upper bound for the 3-connectivity of the Cartesian product of any two connected graphs with orders at least 5. Especially, the exact values for some special cases are determined. Among our results, we also study the pendant tree-connectivity of Cayley graphs on Abelian groups of small degrees and obtain the exact values for τ_{k}(G), where G is a cubic or 4-regular Cayley graph on Abelian groups, 3≤ k≤ n. | |
Keywords: k-connectivity, pendant tree-connectivity, Cartesian product, Cayley graph | |
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