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

A distance-regular graph of diameter 3 with the second eigenvalue $${{\theta }_{1}} = {{a}_{3}}$$
 is called a Shilla graph. For a Shilla graph Γ, the number a = a3 divides k and we set $$b = b(\Gamma ) = k{\text{/}}a$$
 . Three infinite families of Shilla graphs with the following admissible intersection arrays were found earlier: {b(b2 – 1), b2(b – 1), b2; 1, 1, $$({{b}^{2}} - 1)(b - 1)\} $$
 (I.N. Belousov), $$\{ {{b}^{2}}(b - 1){\text{/}}2,(b - 1)({{b}^{2}} - b + 2){\text{/}}2$$
 , $$b(b - 1){\text{/}}4;$$
 $$1,b(b - 1){\text{/}}4,b{{(b - 1)}^{2}}{\text{/}}2\} $$
 (Koolen, Park), and {(s + 1)
 $$({{s}^{3}} - 1),{{s}^{4}},{{s}^{3}};1,{{s}^{2}},s({{s}^{3}} - 1)\} $$
 (Belousov). In this paper, it is proved that, in the first family, there exists a unique graph, namely, a generalized hexagon of order 2, whereas there are no graphs in the second or third families.

Three Infinite Families of Shilla Graphs Do Not Exist

Previously, the authors found the parameters of exceptional strongly regular graphs with eigenvalue 3 (theorem). It is proved that, if Γ is a distance-regular graph whose local subgraphs are strongly regular with parameters listed in items (3)–(10) of the theorem, then Γ is a strongly regular graph, Taylor graph, or a graph with intersection array {100, 66, 1; 1, 33, 100}.

On extensions of small strongly regular graphs with eigenvalue 3

A Shilla graph with Intersection Array {12, 10, 2; 1, 2, 8} Does not Exist

On amply regular locally GQ(4, 6)-graphs

We call a strongly regular graph with $v = \left( {\begin{array}{*{20}c} m \\ 2 \\ \end{array} } \right)$ and k = 2(m − 2) a Higman graph. In Higman graphs, the parameter µ takes values 4, 6, 7, and 8. We find possible orders of automorphisms of Higman graphs with µ = 6 and study the structure of fixed-point subgraphs of these automorphisms.

A. A. Makhnev

Papers

Three Infinite Families of Shilla Graphs Do Not Exist

On extensions of small strongly regular graphs with eigenvalue 3

A Shilla graph with Intersection Array {12, 10, 2; 1, 2, 8} Does not Exist

On amply regular locally GQ(4, 6)-graphs

Automorphisms of Higman graphs with µ = 6