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On a Generalization of the Bipartite Graph $D(k,q)$ | | Xiaoyan Cheng
; Yuansheng Tang
; Huaxiong Wang
; | | Date: |
6 Jul 2017 | | Abstract: | In this paper, we deal with a generalization $Gamma(Omega,q)$ of the
bipartite graphs $D(k,q)$ proposed by Lazebnik and Ustimenko, where $Omega$ is
a set of binary sequences that are adopted to index the entries of the
vertices. A few sufficient conditions on $Omega$ for $Gamma(Omega,q)$ to
admit a variety of automorphisms are proposed. A sufficient condition for
$Gamma(Omega,q)$ to be edge-transitive is proposed further. A lower bound of
the number of the connected components of $Gamma(Omega,q)$ is given by
showing some invariants for the components. For $Gamma(Omega,q)$, paths and
cycles which contain vertices of some specified form are investigated in
details. Some lower bounds for the girth of $Gamma(Omega,q)$ are then shown.
In particular, one can give very simple conditions on the index set $Omega$ so
as to assure the generalized graphs $Gamma(Omega,q)$ to be a family of graphs
with large girth. | | Source: | arXiv, 1707.1633 | | Services: | Forum | Review | PDF | Favorites | |
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