4 research outputs found

    Connectivity and diameter in distance graphs

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    For n∈Nn\in \mathbb{N} and D⊆ND\subseteq \mathbb{N}, the distance graph PnDP_n^D has vertex set {0,1,…,n−1}\{ 0,1,\ldots,n-1\} and edge set {ij∣0≤i,j≤n−1,∣j−i∣∈D}\{ ij\mid 0\leq i,j\leq n-1, |j-i|\in D\}. The class of distance graphs generalizes the important and very well-studied class of circulant graphs which have been proposed for numerous network applications. In view of fault tolerance and delay issues in these applications, the connectivity and diameter of circulant graphs have been studied in great detail. Our main contributions are hardness results concerning computational problems related to the connectivity and diameter of distance graphs and a number-theoretic characterization of the connected distance graphs PnDP_n^D for ∣D∣=2|D|=2

    Diameter of generalized Petersen graphs

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    Due to their broad application to different fields of theory and practice, generalized Petersen graphs GPG(n,s)GPG(n,s) have been extensively investigated. Despite the regularity of generalized Petersen graphs, determining an exact formula for the diameter is still a difficult problem. In their paper, Beenker and Van Lint have proved that if the circulant graph Cn(1,s)C_n(1,s) has diameter dd, then GPG(n,s)GPG(n,s) has diameter at least d+1d+1 and at most d+2d+2. In this paper, we provide necessary and sufficient conditions so that the diameter of GPG(n,s)GPG(n,s) is equal to d+1,d+1, and sufficient conditions so that the diameter of GPG(n,s)GPG(n,s) is equal to d+2.d+2. Afterwards, we give exact values for the diameter of GPG(n,s)GPG(n,s) for almost all cases of nn and s.s. Furthermore, we show that there exists an algorithm computing the diameter of generalized Petersen graphs with running time OO(lognn)
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