Non-bipartite distance-regular graphs with a small smallest eigenvalue

In 2017, Qiao and Koolen showed that for any fixed integer $D\geq 3$, there are only finitely many such graphs with $\theta_{\min}\leq -\alpha k$, where $0<\alpha<1$ is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small $\theta_{\min}$ compared with $k$. In particular, we will show that if $\theta_{\min}$ is relatively close to $-k$, then the odd girth $g$ must be large. Also we will classify the non-bipartite distance-regular graphs with $\theta_{\min} \leq \frac{D-1}{D}$ for $D =4,5$.


Introduction
The odd girth of a non-bipartite graph is the length of its shortest odd cycle. Let Γ be a non-bipartite distance-regular graph with valency k, diameter D, odd girth g and smallest eigenvalue θ min . In [6], Qiao and Koolen showed that for any fixed integer D ≥ 3, there are only finitely many such graphs with θ min ≤ −αk, where 0 < α < 1 is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small θ min compared with k. In the next result, we will show that if θ min is relatively close to −k, then the odd girth g must be large. Theorem 1.1. Let Γ be a non-bipartite distance-regular graph with valency k and odd girth g, having smallest eigenvalue θ min . Then there exists a constant ε(g) > 0 such that θ min ≥ −(1 − ε(g))k.
Remark 1.2. The positive constant ε(g) goes to 0 as the odd girth g goes to ∞. For example, the (2t + 1)-gon has valency k = 2, odd girth g = 2t + 1 and smallest eigenvalue θ min = 2 cos( 2tπ 2t+1 ). Thus, ε(g) ≤ 1 + θ min k = 2 cos 2 ( tπ 2t+1 ). In [6], Qiao and Koolen classified non-bipartite distance-regular graphs with valency k, diameter D ≤ 3 and smallest eigenvalue θ min ≤ −k/2. Using Theorem 1.1, we will classify non-bipartite distance-regular graphs with valency k, diameter D and smallest eigenvalue θ min ≤ − D−1 D k, when D = 4 or 5. This paper is organized as follows. In the next section, we give the definitions and some preliminary results. In section 3, we give a proof of Theorem 1.1. In the last section, we give a proof of Theorem 1.3.

Preliminaries
For more background, see [4] and [7]. All the graphs considered in this paper are finite, undirected and simple. Let Γ be a graph with vertex set V = V (Γ) and edge set E = E(Γ). Denote x ∼ y if the vertices x, y ∈ V are adjacent. The distance d(x, y) = d Γ (x, y) between two vertices x, y ∈ V (Γ) is the length of a shortest path connecting x and y. The maximum distance between two vertices in Γ is the diameter D = D(Γ). We use Γ i (x) for the set of vertices at distance i from x and write, for the sake of simplicity, Γ(x) := Γ 1 (x). The degree of x is the number |Γ(x)| of vertices adjacent to it. A graph is regular with valency k if the degree of each of its vertices is k. The girth and odd girth of a graph is the length of its shortest cycle, and shortest odd cycle, respectively. A graph is called bipartite if it has no odd cycle.
A connected graph Γ with diameter D is called distance-regular if there are integers b i , c i (i = 0, 1, . . . , D) such that for any two vertices x, y ∈ V (Γ) with d(x, y) = i, there are exactly c i neighbors of y in Γ i−1 (x) and b i neighbors of y in Γ i+1 (x), where we define b D = c 0 = 0. In particular, Γ is a regular graph with valency k := b 0 . We define For a distance-regular graph Γ and a vertex x ∈ V (Γ), we denote and hence it does not depend on x. The numbers a i , b i and c i (i = 0, 1, . . . , D) are called the intersection numbers, and the array Let Γ be a distance-regular graph with v vertices and diameter D. Let A i (i = 0, 1, . . . , D) be the (0, 1)-matrix whose rows and columns are indexed by the vertices of Γ and the (x, y)-entry is 1 whenever d(x, y) = i and 0 otherwise. We call A i the distance-i matrix and A := A 1 the adjacency matrix of Γ. The eigenvalues θ 0 > θ 1 > · · · > θ D of the graph Γ are just the eigenvalues of its adjacency matrix A. We denote m i the multiplicity of θ i . Note that the D + 1 distinct eigenvalues of Γ are precisely the eigenvalues of L (see [7,Proposition 2.7]).
For each eigenvalue θ i of Γ, let U i be a matrix with its columns forming an orthonormal basis for the eigenspace associated with θ i . And The set of distance matrices {A 0 = I, A 1 , A 2 , . . . , A D } forms a basis of a commutative R-algebra A, known as the Bose-Mesner algebra. The set of minimal idempotents [4, p.45]), such that the following relations hold . Let E i = U i U T i be the minimal idempotent associated with θ i , where the columns of U i form an orthonormal basis of the eigenspace associated with θ i . We denote the x-th row hence all the vectorsx are unit vectors and the cosine of the angle between two vectorsx andŷ is u j (θ i ) := The map x →x is called a normalized representation and the sequence (u j (θ i )) D j=0 is called the standard sequence of Γ, associated with θ i . As AU i = θ i U i , we have θ ix = y∼xŷ , and hence the following holds: with u 0 (θ i ) = 1 and u 1 (θ i ) = θ i k .
Proof. We only give a proof of Equation (4).

Lemma 2.2. [4, Proposition 4.1.6]
Let Γ be a distance-regular graph with valency k and diameter D. Then the following conditions hold iii) k i 's (i = 1, 2, . . . , D) are positive integers, iv) the multiplicities are positive integers.

Main Theorem
In this section we will prove our main result. Proof of Theorem 1.1 If g = 3, then θ min ≥ − k 2 by [7, Proposition 2.11]. So we may assume g ≥ 5. Let t = g−1 2 and ∆ be a g-gon in Γ. Let (u i ) D i=0 be the standard sequence associated with the smallest eigenvalue θ = θ min .
Now we consider the case c t > ζk.
4 Distance-regular graphs with relatively small θ min In this section we study distance-regular graphs with relatively small θ min . In the rest of this section we will give a proof of Theorem 1.3.