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Spectral Excess Theorem. Let Γ be a connected regular graph with d+1 distinct eigenvalues. Then Γ is distance-regular if and only if the average excess equals the spectral excess.

For short proofs of this theorem we refer the reader to [3, 9]. Note that one can even show that the average excess is at most the spectral excess, and that in [3], a bit stronger result is obtained by using the harmonic mean of the number of vertices minus the excess, instead of the arithmetic mean.

2 The spectral characterization

Let Γ be a connected k-regular graph with adjacency matrix A having d+1 distinct eigenvalues k = λ0 > λ1 > · · · > λd and finite odd-girth at least 2d + 1. It follows that every u has vertices at distance d, because otherwise the vertices at odd distance from u on one hand and the vertices at even distance from u on the other hand, would give a bipartition of the graph, contradicting that the odd-girth is finite. Because Γ has diameter D at most d, it follows that D = d, and that the odd-girth equals 2d + 1. i Because (A )uv counts the number of walks of length i in Γ from u to v, it follows that p(A) has zero diagonal for any odd polynomial p of degree at most 2d − 1. Therefore also tr p(A) = 0. Because the trace of p(A) can also be expressed in terms of the spectrum of Γ, this also shows that the odd-girth condition on Γ is a condition on the spectrum of Γ. In the following, we make frequent use of polynomials. One of these is the Hoffman polynomial H defined by H(x)= n d (x − λ ), where n is the number of vertices and π = d (k − λ ). π0 Qi=1 i 0 Qi=1 i This polynomial satisfies H(A)= J, the all-ones matrix. Let us now consider two arbitrary vertices u, v at distance d. By considering the Hoffman d π0 polynomial, it follows that (A )uv = n . By considering the minimal polynomial (or (x − k)H), d+1 d d d it follows that (A )uv − a˜d(A )uv = 0, wherea ˜d = Pi=0 λi is the coefficient of x in the d+1 π0 minimal polynomial. Hence (A )uv =a ˜d n . n 2d+1 Lemma. The average excess kd of Γ equals 2 tr A . a˜dπ0

Proof. For a vertex u, let Γd(u) be the set of vertices at distance d from u. Then

2d+1 d d+1 2 2 (A )uu = X (A )uv(A )vu = kd(u)˜adπ0/n , v∈Γd(u)

2 2d+1 where kd(u)= |Γd(u)| is the excess of u. Therefore kda˜dπ0/n = tr A anda ˜d 6= 0. ⊔⊓

In order to apply the spectral excess theorem, we have to ensure that kd = pd(k). However, n 2d+1 pd(k) and 2 tr A only depend on the spectrum of Γ, hence so does kd by the lemma. a˜dπ0

2 ′ Therefore, if Γ is cospectral with a distance-regular graph Γ , then the average kd must equal ′ pd(k), because it does so for Γ . Because the spectrum of a graph determines whether it is regular and connected, and determines its odd girth, we hence have: Corollary. (Huang and Liu [11]) Any graph cospectral with a generalized odd graph, is a generalized odd graph.

3 The odd-girth characterization

Now let us show that kd = pd(k) for a connected regular graph Γ having d+1 distinct eigenvalues and finite odd-girth at least 2d+1. To do this, we need some basic properties of the predistance 1 polynomials; see also [9]. First, hp,qi = n tr(p(A)q(A)) defines an inner product (determined by the spectrum of Γ) on the space of polynomials modulo the minimal polynomial of Γ. Using this inner product, one can find an orthogonal system of so-called predistance polynomials pi, i = 0, 1,...,d, where pi has degree i and is normalized such that hpi,pii = pi(k) 6= 0. The predistance polynomials resemble the distance polynomials of a distance-regular graph; they also satisfy a three-term recurrence:

xpi = βi−1pi−1 + αipi + γi+1pi+1, i = 0, 1,...,d, where we let β−1 = 0 and γd+1pd+1 = 0 (the latter we may consider as a multiple of the d minimal polynomial). A final property of these polynomials is that i=0 pi equals the Hoffman P n polynomial H. This implies that the leading coefficient of pd equals π0 (the same as that of H). For the graph Γ under consideration, specific properties hold. It is easy to show by induction that αi = 0 for i < d and that pi is an even or odd polynomial depending on whether i is even or odd, for all i ≤ d. Indeed, it is clear that p0 = 1 is even and p1 = x is odd, and hence that α0 = 0. Now suppose that αi = 0 for i

1 2 n 2d+1 αdpd(k)= hxpd,pdi = tr(Apd(A) )= 2 tr A . n π0

Thus, we have almost shown that this expression for pd(k) and the one for kd in the lemma are the same; what remains is to show that αd =a ˜d. Therefore, consider again vertices u and v at distance d. Then

n d+1 αd = αd(H(A))uv = αd(pd(A))uv = (Apd(A))uv = (A )uv =a ˜d. π0 where the second last step follows because xpd is odd or even, and therefore has no term of degree d. Thus, kd = pd(k) and by the spectral excess theorem we derive that Γ is distance- regular, which finishes the proof of our result.

3 Theorem. Let Γ be a connected regular graph with d+1 distinct eigenvalues and finite odd-girth at least 2d + 1. Then Γ is a distance-regular generalized odd graph.

It is unclear whether we can drop the regularity condition on Γ, or in other words, whether there exist nonregular graphs with d + 1 distinct eigenvalues and odd-girth 2d + 1. For nonregular graphs it matters what matrix we consider (adjacency, Laplacian, etc.). However, for d = 2 we know the following:

Proposition. For the adjacency matrix, as well as for the Laplacian matrix, a connected graph with odd-girth five and three distinct eigenvalues is regular (and hence distance-regular).

Proof. For the adjacency matrix A we consider the minimal polynomial m. Suppose λ0 > λ1 > λ2 are the distinct eigenvalues of A. The diagonal of m(A)= O gives that (λ0 + λ1 + λ2)ku = −λ0λ1λ2, where ku is the valency of vertex u. In case λ0 + λ1 + λ2 = λ0λ1λ2 = 0, it follows that λ0 = −λ2 and λ1 = 0, so the graph would be bipartite, which is false. Thus ku is constant. For a graph whose Laplacian matrix has three distinct eigenvalues it is known that the number µ of common nonneighbors of two adjacent vertices is constant (see [4]). Since there are no triangles, it follows that if u and v are adjacent, then ku + kv = n − µ. This implies that any two vertices at distance two have the same valency. The graph is connected with at least one odd cycle, hence there exists a walk of even length between any two vertices u and v. Because there are no triangles, every even vertex on that walk (which includes u and v) has the same valency. ⊔⊓ For the adjacency matrix we also managed to prove regularity for the analogous cases with four and five distinct eigenvalues, but we choose not to include the technical details.

Acknowledgements The authors thank the referees for their useful comments.

References

[1] A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs. Springer-Verlag, Berlin, 1989.

[2] D.M. Cvetkovi´c, M. Doob, and H. Sachs, Spectra of Graphs, third edition. Johann Ambrosius Barth Verlag, 1995. (First edition: Deutscher Verlag der Wissenschaften, Berlin 1980; Academic Press, New York 1980.)

[3] E.R. van Dam, The spectral excess theorem for distance-regular graphs: a global (over)view. Electron. J. Combin. 15 (2008), no. 1, R129.

[4] E.R. van Dam and W.H. Haemers, Graphs with constant µ and µ. Discrete Math. 182 (1998), 293–307.

[5] E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003), 241-272.

[6] E.R. van Dam and W.H. Haemers, Developments on spectral characterizations of graphs. Discrete Math. 309 (2009), 576–586.

[7] E.R. van Dam, W.H. Haemers, J.H. Koolen, and E. Spence, Characterizing distance-regularity of graphs by the spectrum. J. Combinatorial Th. A 113 (2006), 1805-1820.

4 [8] M.A. Fiol, Algebraic characterizations of distance-regular graphs. Discrete Math. 246 (2002), 111–129.

[9] M.A. Fiol, S. Gago, and E. Garriga, A simple proof of the spectral excess theorem for distance-regular graphs. Linear Algebra Appl. 432 (2010), 2418–2422.

[10] M.A. Fiol and E. Garriga, From local adjacency polynomials to locally pseudo-distance-regular graphs. J. Combinatorial Th. B 71 (1997) 162–183.

[11] T. Huang and C. Liu, Spectral characterization of some generalized odd graphs. Graphs Combin. 15 (1999), 195–209.

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