| Title: | On the uniform structure of bipartite graphs admitting a dual adjacency matrix candidate |
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| Authors: | ID Fernández, Blas (Author)
ID Maleki, Roghayeh (Author)
ID Miklavič, Štefko (Author)
ID Monzillo, Giusy (Author) |
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| Files: |  RAZ_Fernandez_Blas_2026.pdf (318,25 KB)
MD5: 756505BF6B17A60B9FF97ED6DCB72932
 https://link.springer.com/article/10.1007/s10801-026-01546-3
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | FAMNIT - Faculty of Mathematics, Science and Information Technologies
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| Abstract: | 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. |
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| Keywords: | uniform property, dual adjacency matrix, Q-polynomial property |
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| Publication version: | Version of Record |
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| Publication date: | 05.06.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | str. 1-17 |
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| Numbering: | Vol. 63, iss. 4, article no. 60 |
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| PID: | 20.500.12556/RUP-23157  |
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| UDC: | 519.17 |
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| ISSN on article: | 0925-9899 |
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| DOI: | 10.1007/s10801-026-01546-3 |
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| COBISS.SI-ID: | 282133507 |
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| Publication date in RUP: | 18.06.2026 |
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| Views: | 29 |
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| Downloads: | 2 |
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