Iskanje po repozitoriju

Opis: Let Γ denote a finite, bipartite, connected graph with vertex set X. Fix x ∈ X and let ε ≥ 3 denote the eccentricity of x. For mutually distinct scalars {θ ∗ i }ε i=0 define a diagonal matrix A∗ = A∗(θ ∗ 0 , θ ∗ 1 , . . . , θ ∗ ε ) ∈ Mat X (R) as follows: for y ∈ X set (A∗)yy = θ ∗ ∂(x,y), where ∂ denotes the shortest path-length distance function of Γ. We say that A∗ is a dual adjacency matrix candidate of Γ with respect to x if the adjacency matrix A ∈ Mat X (R) of Γ and A∗ satisfy A3 A∗ − A∗ A3 + (β + 1)(A A∗ A2 − A2 A∗ A) = ρ(A A∗ − A∗ A) for some scalars β, ρ ∈ R. In this paper, we investigate when bipartite graphs that admit a dual adjacency matrix candidate also admit a uniform structure (in the sense of Terwilliger [6]). To do that, we first define a weakly uniform structure by slightly relaxing the conditions of uniform structure. The main result of this paper is that Γ admits a dual adjacency matrix candidate with respect to x if and only if Γ admits a weakly uniform structure with respect to x whose parameters satisfy some additional conditions. In particular, for β = 2, the weakly uniform structure is indeed a uniform structure.
Ključne besede: uniform property, dual adjacency matrix, Q-polynomial property
Objavljeno v RUP: 18.06.2026; Ogledov: 271; Prenosov: 8
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