Distance-regular graphs without 4-claws

Abstract

We determine the distance-regular graphs with diameter at least 3 and c 2 ≥ 2 but without induced K 1,4 -subgraphs.

Using Symbolic Computation to Prove Nonexistence of Distance-Regular Graphs

“…sage: drg. DRGParameters ( [6 , 4, 4], [1,1,3]). genPoly parameters () (6,2,2) Note that the existence of a strongly regular graph for which (g, s, t) are defined does not imply the existence of a corresponding generalized quadrangle.…”

“…sage: q7. mergeClasses (2,3,6) Parameters of a distance−regular graph with intersection array {63, 30, 1; 1, 30, 63} Note that mergeClasses (2) gives the parameters of the bipartite half for bipartite graphs, and of the complement for non-antipodal strongly regular graphs.…”

“…where a = a 3 , c = c 2 and p = p 3 33 . Graphs belonging to this family are Q-polynomial with respect to the natural ordering of eigenvalues precisely when the Krein parameter q 3 11 is zero, which is equivalent to…”

“…The proof parallels that of [37,Lems. 1,3] -in fact, a significant part of the proof may be extended to the entire family (3), as it has been done in [59]. The computation needed to obtain the results in this section is illustrated in the jupyter/DRG-d3-2param.ipynb notebook included in the sage-drg package [60].…”

“…8.2.1]. Many Krein parameters are zero, in particular q 3 11 and q 4 11 due to the triangle inequality. We use this fact in the proof of the following statement.…”