On the conjecture for the girth of the bipartite graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>

Cheng, Xiaomin; Chen, Wenbing; Tang, Yizhen

doi:10.1016/j.disc.2016.03.018

Cited by 7 publications

(1 citation statement)

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“…It was also conjectured further in [4] that the girth of D(k, q) is equal to k + 5 for odd k and all q ≥ 4. This conjecture was proved in [19] when (k + 5)/2 is a power of the characteristic of F q and in [20] when (k+5)/2 is the product of a factor of q −1 and a power of the characteristic of F q , respectively. For k ≥ 6, it was shown in [5] that the graph D(k, q) is disconnected and has at least q t−1 components (any two being isomorphic), where t = ⌊(k + 2)/4⌋, by showing some invariants which are fixed in each component.…”

Section: Introductionmentioning

confidence: 92%

On the girth of the bipartite graph D(k,q)

Cheng

Chen

Tang

2014

Discrete Mathematics

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In this paper, we deal with a generalization Γ(Ω, q) of the bipartite graphs D(k, q) proposed by Lazebnik and Ustimenko, where Ω is a set of binary sequences that are adopted to index the entries of the vertices. A few sufficient conditions on Ω for Γ(Ω, q) to admit a variety of automorphisms are proposed. A sufficient condition for Γ(Ω, q) to be edge-transitive is proposed further. A lower bound of the number of the connected components of Γ(Ω, q) is given by showing some invariants for the components. For Γ(Ω, q), paths and cycles which contain vertices of some specified form are investigated in details. Some lower bounds for the girth of Γ(Ω, q) are then shown. In particular, one can give very simple conditions on the index set Ω so as to assure the generalized graphs Γ(Ω, q) to be a family of graphs with large girth.

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