Abstract
In the recently published atlas of graphs [9] the general listing of graphs with diagrams went up to V=7 vertices but the special listing for connected bipartite graphs carried further up to V=8. In this paper we wish to study the random accessibility of these connected bipartite graphs by means of random walks on the graphs using the degree of the “gratis” starting point as a weighting factor. The accessibility is then related to the concept of reliability of the graphs with only edge failures. Exact expectation results for accessibility are given for any complete connected bipartite graph N1 cbp N2 (where “cbp” denotes “connected bipartite”) for several values of J (the number of new vertices searched for). The main conjecture in this paper is that for any complete connected bipartite graph N1 cbp N2: if |N1−N2| ≤ 1, then the graph is both uniformly optimal in reliability and optimal in random accessibility within its family. Numerical results are provided to support the conjecture.
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Ath, Y., Ebneshahrashoob, M., Gao, T. et al. Accessibility and Reliability for Connected Bipartite Graphs. Methodology and Computing in Applied Probability 4, 153–161 (2002). https://doi.org/10.1023/A:1020689507456
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DOI: https://doi.org/10.1023/A:1020689507456