Distance-Regular Graphs

In theoretical medicine, topological indices are defined to test the medicine and pharmacy characteristics, such as melting point, boiling point, toxicity and other biological activities. As basic molecular structures, hexagonal jagged-rectangle and distance-regular structure are widely appeared in medicine, pharmacy and biology engineering. In this paper, we study the chemical properties of hexagonal jagged-rectangle from the mathematical point of view. Several vertex distance-based indices are determined. Furthermore, the Wiener related indices of distance-regular structure are also considered.

Characteristics studies of molecular structures in drugs.

Strongly Regular Graphs and Partial Geometries

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.

Distance-regular graphs

Pseudo-skeleton approximations with better accuracy estimates

On graphs with strongly regular local subgraphs with parameters (111, 30, 5, 9) or (169, 42, 5, 12)

A geometry of rank 2 is an incidence system (P, $\mathcal{B}$), where P is a set of points and $\mathcal{B}$ is a set of subsets from P, called blocks. Two points are called collinear if they lie in a common block. A pair (a, B) from (P, $\mathcal{B}$) is called a flag if its point belongs to the block, and an antiflag otherwise. A geometry is called φ-uniform (φ is a natural number) if for any antiflag (a, B) the number of points in the block B collinear to the point a equals 0 or φ, and strongly φ-uniform if this number equals φ. In this paper, we study φ-uniform extensions of partial geometries pGα(s, t) with φ = s and strongly φ-uniform geometries with φ = s − 1. In particular, the results on extensions of generalized quadrangles, obtained earlier by Cameron and Fisher, are extended to the case of partial geometries.

Uniform extensions of partial geometries

A. A. Makhnev and M. S. Nirova found the intersection arrays of distance regular graphs with $$\lambda=2$$
 and at most 4096 vertices. For graphs of diameter $$4$$
 , of most interest is the array $$\{21,18,12,4;1,1,6,21\}$$
 in this list. In this paper, we find the possible orders and fixed point subgraphs of the automorphisms of a distance regular graph with intersection array $$\{21,18,12,4;1,1,6,21\}$$
 .

Automorphisms of a Distance Regular Graph with Intersection Array $$\{21,18,12,4;1,1,6,21\}$$

We point out possible automorphisms of a distance-regular graph Γ with intersection array {55, 54, 2; 1, 1, 54} and spectrum 551, 71617,−1110,−81408.

The Largest Moore Graph and a Distance-Regular Graph with Intersection Array {55, 54, 2; 1, 1, 54}

Let Γ be a distance regular graph with intersection array {35, 32, 1; 1, 4, 35} and let G = Aut(Γ) act transitively on the set of vertices of the graph Γ. It is shown that G is a {2, 3}-group.

A. A. Makhnev

Papers

On graphs with strongly regular local subgraphs with parameters (111, 30, 5, 9) or (169, 42, 5, 12)

Uniform extensions of partial geometries

Automorphisms of a Distance Regular Graph with Intersection Array $$\{21,18,12,4;1,1,6,21\}$$

The Largest Moore Graph and a Distance-Regular Graph with Intersection Array {55, 54, 2; 1, 1, 54}

Automorphism Group of a Distance-Regular Graph with Intersection Array {35 , 32 , 1; 1 , 4 , 35}