Geometric distance-regular graphs without 4-claws

Abstract

A non-complete distance-regular graph is called geometric if there exists a set C of Delsarte cliques such that each edge of lies in a unique clique in C. In this paper we determine the non-complete distance-regular graphs satisfying max 3, 8 3 (a 1 + 1) < k < 4a 1 + 10 − 6c 2 . To prove this result, we first show by considering nonexistence of 4-claws that any non-complete distance-regular graph satisfying max 3, 8 3 (a 1 + 1) < k < 4a 1 + 10 − 6c 2 is a geometric distance-regular graph with smallest eigenvalue −3. Moreover, we classify the geometric distance-regular graphs with smallest eigenvalue −3. As an application, two feasible intersection arrays in the list of [7, Chapter 14] are ruled out.

On the automorphism groups of distance-regular graphs and rank-4 primitive coherent configurations

“…For the case of distance-regular graphs of diameter 3 we use two recent classification results for geometric distance-regular graphs with smallest eigenvalue −3 proven by Bang [11] and Bang, Koolen [12]. Since Bang's classification result includes a lot of cases we state here the short corollary of her result that fits our purposes.…”

“…In the remaining case, we use Metsch's criteria for geometricity (see Theorem 3.25) to show that X is a geometric distance-regular graph with smallest eigenvalue −m, where m is bounded above by a function of the diameter. In the case of diameter 3 we obtain a tight bound on m, reducing the problem to geometric distance-regular graphs with smallest eigenvalue −3 and we use the almost complete classification of such graphs obtained by Bang [11] and Bang, Koolen [12] (see Theorem 3.34 in this paper). No analogous classification for the geometric distance-regular graphs with smallest eigenvalue −m appears to be known for m ≥ 4.…”

On the spectral gap and the automorphism group of distance-regular graphs

“…Bang and Koolen [17] proved that all but finitely many distance-regular graphs with smallest eigenvalue −m and µ ≥ 2 are geometric. For geometric distance-regular graphs with smallest eigenvalue ≥ −3 and µ ≥ 2 Bang [6] and Bang, Koolen [7] gave a complete classification. Moreover, they conjectured [17,Conjecture 7.4] that for any integer m all but finitely many geometric distance-regular graphs with smallest eigenvalue −m and µ ≥ 2 are known.…”

Дистанционно Регулярный Граф С Массивом Пересечений $\{45,30,7;1,2,27\}$ Не Существует

“…В работе [5] изучались геометрические дистанционно регулярные графы с наименьшим собственным значением −3 (следовательно, не содержащие 4-лап). В [5, теорема 4.3] все графы с таким свойством перечислены в 12-и классах (в каждом из которых с точностью до некоторых параметров указан массив пересечений).…”

“…В одном из этих классов все графы имеют диаметр три, а массив пересечений имеет вид {3α+3, 2α+2, α+2−β; 1, 2, 3β}, где α β 1 целые числа. Из списка [1] допустимых массивов пересечений дистанционно регулярных графов (с небольшим числом вершин) в этот класс попадают только массивы графов Хэмминга H(3, α+2) и графа Дуба диаметра 3 (при β = 1), и массив пересечений {45, 30, 7; 1, 2, 27} (при α = 14, β = 9), для которого существование соответствующего графа, как отмечает автор [5], неизвестно.…”