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Incidence graphs constructed from \(t\)-designs. (English) Zbl 1461.05136

Summary: Let \(D\) be a non-trivial simple \(t\)-design. In this paper, for \(1 \le s \le t\), we generalize the concept of the incidence graph of \(D\) and construct a new bi-partite regular graph \(\Gamma\). We obtain that the edge-transitivity of the graph \(\Gamma\) is equivalent to the \(s\)-ag-transitivity of the design \(D\). We then, for \(s = 2\), classify the semisymmetric graphs among the graphs \(\Gamma\) constructed from bi-planes and triplanes. Finally, we study the connectedness and the energy of incidence graphs. Several open problems are proposed, one of which asks whether the incidence graphs have large vertex-connectivity.

MSC:

05C51 Graph designs and isomorphic decomposition
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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