Homogeneous Graphs and Regular Near Polygons

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“…In this section, we show the concept of tight is closely related to the concept of 1-homogeneous that appears in the work of Nomura [14][15][16]. …”

“…Our third characterization of the tight condition involves the concept of 1-homogeneous that appears in the work of Nomura [14][15][16]. See also Curtin [7].…”

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“…We study 1-homogeneous graphs in the sense of Nomura [16] (defined later in this section). Some examples of such graphs are distance-regular graphs with at most one i, such that a i = 0 (e.g.…”

“…For y ∈ i (x) and integers j and h we define D h j (x, y) = j (x) ∩ h (y) and p i jh (x, y) = |D h j (x, y)|. Then is i-homogeneous in the sense of Nomura [16] when the distance partition corresponding to any pair x, y of vertices at distance i, i.e., the collection of nonempty sets D j h (x, y), is an equitable partition, and the parameters corresponding to all such equitable partitions are independent of vertices x and y at distance i. Note that the graph is 0-homogeneous if and only if it is distance-regular, and that if is 1-homogeneous then it is distance-regular.…”

“…(a) the distance partition of 1-homogeneous graph corresponding to two adjacent vertices; (b) the distance partition of the complement of 1-homogeneous graph , corresponding to two vertices at distance 2, where A tower of graphs with their distance partitions corresponding to two adjacent vertices (all but the last one are 1-homogeneous graphs): (a) the Gosset graph is a unique distance-regular graph with intersection array {27, 10, 1; 1, 10, 27}, an antipodal 2-cover of the complete graph K 28 , and it is locally Schläfli graph see [3, pp. 103, 313]; (b) the Schläfli graph is a unique strongly regular graph (27,16,10,8) and it is locally halved 5-cube, see [3, p. 103]; (c) the halved 5-cube, also known as the Clebsch graph, is a unique strongly regular graph (16,10,6,6) and it is locally J (5, 2), i.e., the complement of the Petersen graph, see [3, p. 264] (so the local graph is not 1-homogeneous), the Johnson graph J (5, 2) is a unique strongly regular graph (10,6,3,4) and is locally the 3-prism; (d) the 3-prism has two different distance partitions corresponding to an edge.…”