1-homogeneous, pseudo-1-homogeneous, and 1-thin distance-regular graphs

Abstract

Let denote a distance-regular graph with diameter d 2, and fix a vertex x of . is said to be 1-homogeneous (resp. pseudo-1-homogeneous) with respect to x whenever for all integers h and i between 0 and d, inclusive (resp. for all integers h between 0 and d − 1 and i between 0 and d,

Algebraic Characterizations of Graph Regularity Conditions

“…This algebraic characterization is studied in greater detail for the 1-homogeneous distance-regular graphs in [6] and for the 2-homogeneous bipartite distance-regular graphs in [5].…”

Distance-regular graphs

“…Jurišić and Terwilliger [385] showed among other results that if a 1 = 0 then the edge xy is tight with respect to two distinct real numbers if and only if Γ is pseudo 1-homogeneous with respect to xy and the induced subgraph on Γ 1,1 (x, y) is not a clique. Under the condition a 1 = 0, Curtin and Nomura [157] characterized the situation where Γ is 1-thin with respect to x with precisely two non-isomorphic irreducible T(x)-modules with endpoint one, in terms of the pseudo 1-homogeneous property 21 of the edges xy (y ∈ Γ(x)). They studied in detail the case where a 1 = 0 as well.…”

“…It should be remarked that Curtin and Nomura[157] do not require the existence of the parameter p D,D−1;r,s with respect to x, y 22. See also[591] for a generalization of this result (as well as Nomura's classification[517,518] of bipartite or almost bipartite 2-homogeneous distance-regular graphs) to triangle-free distance-regular graphs.…”

Pseudo 1-homogeneous distance-regular graphs

“…First, we show that the graph is pseudo 1-homogeneous with respect to the edge xy if and only if the edge xy is tight with respect to two distinct real numbers. For the second, we first define the graph to be pseudo 1-homogeneous with respect to a vertex x whenever it is pseudo 1-homogeneous with respect to all edges incident with x. Curtin and Nomura [3] have characterized graphs with this property in terms of their subconstituent algebra. We offer another characterization: the distance-regular graph is pseudo 1-homogeneous with respect to a vertex x such that the local graph of x is connected if and only if allows the above parameterization with d + 1 parameters σ 1 , .…”