Distance-regular graph

Graph families defined by their automorphisms
distance-transitive	→	distance-regular	←	strongly regular
↓
symmetric (arc-transitive)	←	t-transitive,t ≥ 2		skew-symmetric
↓
(if connected) vertex- and edge-transitive	→	edge-transitive and regular	→	edge-transitive
↓		↓		↓
vertex-transitive	→	regular	→	(if bipartite) biregular
↑
Cayley graph	←	zero-symmetric		asymmetric

In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and the distance between v and w.

Some authors exclude the complete graphs and disconnected graphs from this definition.

Every distance-transitive graph is distance-regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group.

Intersection arrays

The intersection array of a distance-regular graph is the array ${\displaystyle (b_{0},b_{1},\ldots ,b_{d-1};c_{1},\ldots ,c_{d})}$ in which ${\displaystyle d}$ is the diameter of the graph and for each ${\displaystyle 1\leq j\leq d}$, ${\displaystyle b_{j}}$ gives the number of neighbours of ${\displaystyle u}$ at distance ${\displaystyle j+1}$ from ${\displaystyle v}$ and ${\displaystyle c_{j}}$ gives the number of neighbours of ${\displaystyle u}$ at distance ${\displaystyle j-1}$ from ${\displaystyle v}$ for any pair of vertices ${\displaystyle u}$ and ${\displaystyle v}$ at distance ${\displaystyle j}$. There is also the number ${\displaystyle a_{j}}$ that gives the number of neighbours of ${\displaystyle u}$ at distance ${\displaystyle j}$ from ${\displaystyle v}$. The numbers ${\displaystyle a_{j},b_{j},c_{j}}$ are called the intersection numbers of the graph. They satisfy the equation ${\displaystyle a_{j}+b_{j}+c_{j}=k,}$ where ${\displaystyle k=b_{0}}$ is the valency, i.e., the number of neighbours, of any vertex.

It turns out that a graph ${\displaystyle G}$ of diameter ${\displaystyle d}$ is distance regular if and only if it has an intersection array in the preceding sense.

Cospectral and disconnected distance-regular graphs

A pair of connected distance-regular graphs are cospectral if their adjacency matrices have the same spectrum. This is equivalent to their having the same intersection array.

A distance-regular graph is disconnected if and only if it is a disjoint union of cospectral distance-regular graphs.

Properties

Suppose ${\displaystyle G}$ is a connected distance-regular graph of valency ${\displaystyle k}$ with intersection array ${\displaystyle (b_{0},b_{1},\ldots ,b_{d-1};c_{1},\ldots ,c_{d})}$. For each ${\displaystyle 0\leq j\leq d,}$ let ${\displaystyle k_{j}}$ denote the number of vertices at distance ${\displaystyle k}$ from any given vertex and let ${\displaystyle G_{j}}$ denote the ${\displaystyle k_{j}}$-regular graph with adjacency matrix ${\displaystyle A_{j}}$ formed by relating pairs of vertices on ${\displaystyle G}$ at distance ${\displaystyle j}$.

Graph-theoretic properties

• ${\displaystyle {\frac {k_{j+1}}{k_{j}}}={\frac {b_{j}}{c_{j+1}}}}$ for all ${\displaystyle 0\leq j<d}$.
• ${\displaystyle b_{0}>b_{1}\geq \cdots \geq b_{d-1}>0}$ and ${\displaystyle 1=c_{1}\leq \cdots \leq c_{d}\leq b_{0}}$.

Spectral properties

• ${\displaystyle G}$ has ${\displaystyle d+1}$ distinct eigenvalues.
• The only simple eigenvalue of ${\displaystyle G}$ is ${\displaystyle k,}$ or both ${\displaystyle k}$ and ${\displaystyle -k}$ if ${\displaystyle G}$ is bipartite.
• ${\displaystyle k\leq {\frac {1}{2}}(m-1)(m+2)}$ for any eigenvalue multiplicity ${\displaystyle m>1}$ of ${\displaystyle G,}$ unless ${\displaystyle G}$ is a complete multipartite graph.
• ${\displaystyle d\leq 3m-4}$ for any eigenvalue multiplicity ${\displaystyle m>1}$ of ${\displaystyle G,}$ unless ${\displaystyle G}$ is a cycle graph or a complete multipartite graph.

If ${\displaystyle G}$ is strongly regular, then ${\displaystyle n\leq 4m-1}$ and ${\displaystyle k\leq 2m-1}$.

Examples

The degree 7 Klein graph and associated map embedded in an orientable surface of genus 3. This graph is distance regular with intersection array {7,4,1;1,2,7} and automorphism group PGL(2,7).

Some first examples of distance-regular graphs include:

• The complete graphs.
• The cycle graphs.
• The odd graphs.
• The Moore graphs.
• The collinearity graph of a regular near polygon.
• The Wells graph and the Sylvester graph.
• Strongly regular graphs of diameter ${\displaystyle 2}$.

Classification of distance-regular graphs

There are only finitely many distinct connected distance-regular graphs of any given valency ${\displaystyle k>2}$. ^[1]

Similarly, there are only finitely many distinct connected distance-regular graphs with any given eigenvalue multiplicity ${\displaystyle m>2}$ ^[2] (with the exception of the complete multipartite graphs).

Cubic distance-regular graphs

The cubic distance-regular graphs have been completely classified.

The 13 distinct cubic distance-regular graphs are K₄ (or Tetrahedral graph), K_3,3, the Petersen graph, the Cubical graph, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the Dodecahedral graph, the Desargues graph, Tutte 12-cage, the Biggs–Smith graph, and the Foster graph.

References

1. Bang, S.; Dubickas, A.; Koolen, J. H.; Moulton, V. (2015-01-10). "There are only finitely many distance-regular graphs of fixed valency greater than two". Advances in Mathematics . 269 (Supplement C): 1–55. arXiv: 0909.5253 . doi: 10.1016/j.aim.2014.09.025 . S2CID   18869283.
2. Godsil, C. D. (1988-12-01). "Bounding the diameter of distance-regular graphs". Combinatorica . 8 (4): 333–343. doi:10.1007/BF02189090. ISSN   0209-9683. S2CID   206813795.

Further reading

• Godsil, C. D. (1993). Algebraic Combinatorics. Chapman and Hall Mathematics Series. New York: Chapman and Hall. ISBN   978-0-412-04131-0. MR   1220704.