Dam, E.R. van

Haemers, W.H.

English

Research Papers in Economics

This is a postprint of:
An odd characterization of the generalized odd graphs
E.R. van Dam & W.H. Haemers
Journal of Combinatorial Theory, B, 2011, vol. 101, nr. 6, pp. 486-489
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http://dbiref.uvt.nl/iPort?request=full_record&db=wo&language=eng&query=doc_id=4578529An odd characterization of the generalized odd graphs¤
Edwin R. van Dam and Willem H. Haemers
Tilburg University, Dept. Econometrics and O.R.,
P.O. Box 90153, 5000 LE Tilburg, The Netherlands,
e-mail: Edwin.vanDam@uvt.nl, Haemers@uvt.nl
Abstract We show that any connected regular graph with d + 1 distinct eigenvalues and odd-girth 2d + 1 is
distance-regular, and in particular that it is a generalized odd graph.
2010 Mathematics Subject Classi¯cation: 05E30, 05C50
Keywords: distance-regular graphs, generalized odd graphs, odd-girth, spectra of graphs, spectral excess theo-
rem, spectral characterization
1 Introduction
The odd-girth of a graph is the length of the shortest odd cycle. A generalized odd graph
is a distance-regular graph of diameter D and odd girth 2D + 1. It is also called an almost-
bipartite distance-regular graph, or a regular thin near (2D + 1)-gon. Well-known examples
of such graphs are the Odd graphs (also known as the Kneser graphs K(2D + 1;D)), and the
folded (2D + 1)-cubes.
In this note, we shall characterize these graphs, by showing that any connected regular graph
with d+1 distinct eigenvalues and odd-girth (at least) 2d+1 is a distance-regular generalized
odd graph. We remark that D = d for distance-regular graphs, but for arbitrary connected
graphs we only have the inequality D · d. In general it is not true that any connected regular
graph with diameter D and odd-girth 2D+1 is a generalized odd graph. Counterexamples can
easily be found for the case D = 2 (among the triangle-free regular graphs with diameter two
there are many graphs that are not strongly regular).
Huang and Liu [11] proved that any graph with the same spectrum as a generalized odd
graph is such a graph. Because the odd-girth of a graph follows from the spectrum, our
characterization is a generalization of this result.
For background on distance-regular graphs we refer the reader to [1], for eigenvalues of
graphs to [2], for spectral characterizations of graphs to [5, 6], and for spectral and other
algebraic characterizations of distance-regular graphs to [7] and [8], respectively. To show the
claimed characterization, we shall use the so-called spectral excess theorem due to Fiol and
Garriga [10]. Let ¡ be a connected k-regular graph with d + 1 distinct eigenvalues. The
excess of a vertex u of ¡ is the number of vertices at distance d from u. We also need the
¤This version is published in Journal of Combinatorial Theory, Series B 101 (2011), 486-489.
1so-called predistance polynomial pd of ¡, which will be explained in some detail in Section 3.
The important property of pd is that the value of pd(k) | the so-called spectral excess |
only depends on the spectrum of ¡ (in fact, all predistance polynomials depend only on the
spectrum).
Spectral Excess Theorem. Let ¡ be a connected regular graph with d+1 distinct eigenvalues.
Then ¡ 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 ¡ be a connected k-regular graph with adjacency matrix A having d+1 distinct eigenvalues
k = ¸0 > ¸1 > ¢¢¢ > ¸d and ¯nite odd-girth at least 2d + 1. It follows that every vertex 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 ¯nite. Because ¡ has diameter D at most d, it follows that
D = d, and that the odd-girth equals 2d + 1.
Because (Ai)uv counts the number of walks of length i in ¡ 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
trp(A) = 0. Because the trace of p(A) can also be expressed in terms of the spectrum of ¡,
this also shows that the odd-girth condition on ¡ is a condition on the spectrum of ¡. In the
following, we make frequent use of polynomials. One of these is the Ho®man polynomial H
de¯ned by H(x) = n
¼0
Qd
i=1(x ¡ ¸i), where n is the number of vertices and ¼0 =
Qd
i=1(k ¡ ¸i).
This polynomial satis¯es H(A) = J, the all-ones matrix.
Let us now consider two arbitrary vertices u;v at distance d. By considering the Ho®man
polynomial, it follows that (Ad)uv = ¼0
n . By considering the minimal polynomial (or (x¡k)H),
it follows that (Ad+1)uv ¡ ~ ad(Ad)uv = 0, where ~ ad =
Pd
i=0 ¸i is the coe±cient of xd in the
minimal polynomial. Hence (Ad+1)uv = ~ ad
¼0
n .
Lemma. The average excess kd of ¡ equals n
~ ad¼2
0
trA2d+1.
Proof. For a vertex u, let ¡d(u) be the set of vertices at distance d from u. Then
(A2d+1)uu =
X
v2¡d(u)
(Ad)uv(Ad+1)vu = kd(u)~ ad¼2
0=n2;
where kd(u) = j¡d(u)j is the excess of u. Therefore kd~ ad¼2
0=n = trA2d+1 and ~ ad 6= 0. t u
In order to apply the spectral excess theorem, we have to ensure that kd = pd(k). However,
pd(k) and n
~ ad¼2
0
trA2d+1 only depend on the spectrum of ¡, hence so does kd by the lemma.
2Therefore, if ¡ is cospectral with a distance-regular graph ¡0, then the average kd must equal
pd(k), because it does so for ¡0. 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 ¡ having d+1 distinct eigenvalues
and ¯nite odd-girth at least 2d+1. To do this, we need some basic properties of the predistance
polynomials; see also [9]. First, hp;qi = 1
n tr(p(A)q(A)) de¯nes an inner product (determined
by the spectrum of ¡) on the space of polynomials modulo the minimal polynomial of ¡. Using
this inner product, one can ¯nd 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 = ¯i¡1pi¡1 + ®ipi + °i+1pi+1; i = 0;1;:::;d;
where we let ¯¡1 = 0 and °d+1pd+1 = 0 (the latter we may consider as a multiple of the
minimal polynomial). A ¯nal property of these polynomials is that
Pd
i=0 pi equals the Ho®man
polynomial H. This implies that the leading coe±cient of pd equals n
¼0 (the same as that of
H).
For the graph ¡ under consideration, speci¯c properties hold. It is easy to show by induction
that ®i = 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
®0 = 0. Now suppose that ®i = 0 for i < j < d and that pi is even or odd (depending on i) for
i · j. Then the three-term recurrence implies that ®jpj(k) = hxpj;pji = 1
n tr(Apj(A)2) = 0
because xp2
j is an odd polynomial of degree at most 2d ¡ 1. Hence ®j = 0 and then it follows
from the recurrence that pj+1 is even or odd, which ¯nishes the inductive argument.
What we shall use now is that xp2
d is an odd polynomial. It follows that
®dpd(k) = hxpd;pdi =
1
n
tr(Apd(A)2) =
n
¼2
0
trA2d+1:
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 ®d = ~ ad. Therefore, consider again vertices u and v at
distance d. Then
®d = ®d(H(A))uv = ®d(pd(A))uv = (Apd(A))uv =
n
¼0
(Ad+1)uv = ~ ad:
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 ¡ is distance-
regular, which ¯nishes the proof of our result.
3Theorem. Let ¡ be a connected regular graph with d+1 distinct eigenvalues and ¯nite odd-girth
at least 2d + 1. Then ¡ is a distance-regular generalized odd graph.
It is unclear whether we can drop the regularity condition on ¡, 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 ¯ve and three distinct eigenvalues is regular (and hence distance-regular).
Proof. For the adjacency matrix A we consider the minimal polynomial m. Suppose ¸0 > ¸1 >
¸2 are the distinct eigenvalues of A. The diagonal of m(A) = O gives that (¸0 + ¸1 + ¸2)ku =
¡¸0¸1¸2, where ku is the valency of vertex u. In case ¸0 + ¸1 + ¸2 = ¸0¸1¸2 = 0, it follows
that ¸0 = ¡¸2 and ¸1 = 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 ¹ 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 ¡ ¹. 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. t u
For the adjacency matrix we also managed to prove regularity for the analogous cases with four
and ¯ve distinct eigenvalues, but we choose not to include the technical details.
Acknowledgements The authors thank the referees for their useful comments.
References
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An odd characterization of the generalized odd graphs.

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