✦ LIBER ✦

Spin Models on Bipartite Distance-Regular Graphs

✍ Scribed by K. Nomura

Publisher: Elsevier Science
Year: 1995
Tongue: English
Weight: 430 KB
Volume: 64
Category: Article
ISSN: 0095-8956
DOI: 10.1006/jctb.1995.1037

No coin nor oath required. For personal study only.

✦ Synopsis

Spin models were introduced by V. Jones (Pac. J. Math. 137 (1989), 311-336) to construct invariants of knots and links. A spin model will be defined as a pair (S=(X, w)) of a finite set (X) and a function (w) on (X \times X) satisfying several axioms. Some important spin models can be constructed on a distance-regular graph (\Gamma=) ((X, E)) with suitable complex numbers (t_{0}, t_{1}, \ldots, t_{d}(d) is the diameter of (\Gamma) ) by putting (w(a, b)=t_{\hat{c}(a, b)}). In this paper we determine bipartite distance-regular graphs which give spin models in this way with distinct (t_{1}, \ldots, t_{d}). We show that such a bipartite distance-regular graph satisfies a strong regularity condition (it is 2-homogeneous), and we classify bipartite distance-regular graphs which satisfy this regularity condition. i 1995 Academic Press, Inc.

📜 SIMILAR VOLUMES

Some Formulas for Spin Models on Distanc

Some Formulas for Spin Models on Distance-Regular Graphs

✍ Brian Curtin; Kazumasa Nomura 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 213 KB

A spin model is a square matrix W satisfying certain conditions which ensure that it yields an invariant of knots and links via a statistical mechanical construction of V. F. R. Jones. Recently F. Jaeger gave a topological construction for each spin model W of an association scheme which contains W

Tails of Bipartite Distance-regular Grap

Tails of Bipartite Distance-regular Graphs

✍ Michael S. Lang 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 205 KB

Let denote a bipartite distance-regular graph with diameter D ≥ 4 and valency k ≥ 3. Let θ 0 > θ 1 > • • • > θ D denote the eigenvalues of and let E 0 , E 1 , . . . , E D denote the associated primitive idempotents. Fix s (1 ≤ s ≤ D -1) and abbreviate E := E s . We say E is a tail whenever the entry

Dual Bipartite Q-polynomial Distance-reg

Dual Bipartite Q-polynomial Distance-regular Graphs

✍ Garth A. Dickie; Paul M. Terwilliger 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 269 KB

Almost 2-homogeneous Bipartite Distance-

Almost 2-homogeneous Bipartite Distance-regular Graphs

✍ Brian Curtin 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 170 KB

Bipartite Q-Polynomial Quotients of Anti

Bipartite Q-Polynomial Quotients of Antipodal Distance-Regular Graphs

✍ John S. Caughman IV 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 91 KB

Let 1 denote a bipartite Q-polynomial distance-regular graph with diameter D 4. We show that 1 is the quotient of an antipodal distance-regular graph if and only if one of the following holds. (i) 1 is a cycle of even length. (ii) 1 is the quotient of the 2D-cube. 1999 Academic Press \* , ..., %\

The Local Structure of a Bipartite Dista

The Local Structure of a Bipartite Distance-regular Graph

✍ Brian Curtin 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 285 KB

In this paper, we consider a bipartite distance-regular graph = (X, E) with diameter d ≥ 3. We investigate the local structure of , focusing on those vertices with distance at most 2 from a given vertex x. To do this, we consider a subalgebra R = R(x) of Mat X (C), where X denotes the set of vertice