Distance-Regular Graphs withbt=1 and Antipodal Double-Covers

Abstract

Let 1 be a distance-regular graph of diameter d and valency k>2. If b t =1 and 2t d, then 1 is an antipodal double-cover. Consequently, if f >2 is the multiplicity of an eigenvalue of the adjacency matrix of 1 and if 1 is not an antipodal doublecover then d 2f&3. This result is an improvement of Godsil's bound. 1996Academic Press, Inc.

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“…In [1], we have already obtained the special case of the main theorem of this paper, i.e., a distance-regular graph of b t = 1, d ≥ 2t and valency k > 2 is an antipodal double cover, which is one of important facts to prove our theorem.…”

“…A (d, 1)-box is called a box that was a key to prove the theorem in [1]. Notice that there are many boxes in an antipodal distance-regular graph Γ of diameter d ≥ 3; namely, given u, y with ∂(u, y) = d, there is a one to one correspondence between v ∈ Γ 1 (u) and x ∈ Γ 1 (y) such that (u, v, x, y) is a box.…”

“…One of the characterization of antipodal distance-regular graphs is that they have a box. These configurations are useful tools when we investigate if a graph is antipodal or not as we see § 2, [1] or [2]. Readers who are familiar with distance distribution diagrams may read some of proofs easily, however, we can do without diagrams.…”

Distance-regular graphs

“…Therefore, a distance-regular graph is said to have classical parameters (D, b, α, β) if its intersection numbers can be expressed as in (7). We note that the parameter b must be an integer not equal to 0 or −1.…”

An Application of a Construction Theory of Strongly Closed Subgraphs in a Distance-regular Graph

“…More information about (h, j)-boxes is given in [1]. In this paper, we say 'a box' for a (2r + 1, r + 1)-box for short.…”