✦ LIBER ✦

The Local Structure of a Bipartite Distance-regular Graph

✍ Scribed by Brian Curtin

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 285 KB
Volume: 20
Category: Article
ISSN: 0195-6698
DOI: 10.1006/eujc.1999.0307

No coin nor oath required. For personal study only.

✦ Synopsis

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 vertices in X at distance 2 from x. R is generated by matrices Ã, J , and D defined as follows. For all y, z ∈ X , the (y, z)-entry of Ã is 1 if y, z are at distance 2, and 0 otherwise. The (y, z)-entry of J equals 1, and the (y, z)-entry of D equals the number of vertices of X adjacent to each of x, y, and z.

We show that R is commutative and semisimple, with dimension at least 2. We assume that dim R is one of 2, 3, or 4, and explore the combinatorial implications of this. We are motivated by the fact that if has a Q-polynomial structure, then dim R ≤ 4.

📜 SIMILAR VOLUMES

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

The Terwilliger Algebra of a 2-Homogeneo

The Terwilliger Algebra of a 2-Homogeneous Bipartite Distance-Regular Graph

✍ Brian Curtin 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 200 KB

Let 1 denote a 2-homogeneous bipartite distance-regular graph with diameter D 3 and valency k 3. Assume that 1 is not isomorphic to a Hamming cube. Fix a vertex x of 1, and let T=T(x) denote the Terwilliger algebra of T with respect to x. We give three sets of generators for T, two of which satisfy

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 \* , ..., %\

Bipartite Distance-Regular Graphs with a

Bipartite Distance-Regular Graphs with an Eigenvalue of Multiplicity itk m

✍ Norio Yamazaki 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 194 KB

We show that, if a bipartite distance-regular graph of valency k has an eigenvalue of multiplicity k, then it becomes 2-homogeneous. Combined with a result on bipartite 2-homogeneous distance-regular graphs by K. Nomura, we have a classification of such graphs.

The Girth of a Directed Distance-Regular

The Girth of a Directed Distance-Regular Graph

✍ D.A. Leonard; K. Nomura 📂 Article 📅 1993 🏛 Elsevier Science 🌐 English ⚖ 149 KB

A note on bipartite subgraphs of triangl

A note on bipartite subgraphs of triangle-free regular graphs

✍ S. C. Locke 📂 Article 📅 1990 🏛 John Wiley and Sons 🌐 English ⚖ 130 KB 👁 2 views

## Abstract Lower bounds on the size of a maximum bipartite subgraph of a triangle‐free __r__‐regular graph are presented.