Dam, E.R. van

English

Research Papers in Economics

Linear Algebra and its Applications 426 (2007) 454–457
www.elsevier.com/locate/laa
Graphs with given diameter maximizing
the spectral radius
E.R. van Dam
Tilburg University, Department of Econometrics and Operations Research, P.O. Box 90153,
5000 LE Tilburg, The Netherlands
Received 31 October 2006; accepted 17 May 2007
Available online 29 May 2007
Submitted by R.A. Brualdi
Abstract
We determine the graphs with maximal spectral radius among the ones on n nodes with diameter D.
© 2007 Elsevier Inc. All rights reserved.
AMS classiﬁcation: 05C50; 05E99; 94C15
Keywords: Graphs; Spectral radius; Diameter
In this note we determine the graphs with maximal spectral radius among the ones on n nodes
with diameter D. This complements the results in [2] on graphs with minimal spectral radius for
given number of nodes and diameter. Motivation for this problem comes from virus spreading in
networks, cf. [2]. Other related work is done by Guo and Shao [5] who determined the trees with
maximal spectral radius for ﬁxed diameter.
Let M(n,D) be the graph obtained from a complete graph on n − D + 2 nodes by removing
anedge,addingapendantpathof
 D
2
 
− 1edgestooneendnodeoftheremovededge,andadding
a pendant path of
 D
2
 
− 1 edges to its other endnode.
Theorem. Let n and D be integers, with 1 <D<n .Then the graph M(n,D) is the graph with
maximal spectral radius among all graphs on n nodes with diameter D.
E-mail address: Edwin.vanDam@uvt.nl
0024-3795/$ - see front matter ( 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.laa.2007.05.026E.R. van Dam / Linear Algebra and its Applications 426 (2007) 454–457 455
The proof of this result will be established in a number of steps. First of all, we need the fact
that adding edges to a connected graph increases the spectral radius (cf. [1, p. 19]). This allows
us to prove the following.
Lemma 1. LetnandD beintegers,with1 <D<n .Thenagraphwithmaximalspectralradius
among all graphs on n nodes with diameter D is a graph obtained from the path PD+1 by
replacing the nodes by cliques, such that nodes in distinct cliques are adjacent if and only if the
corresponding original nodes in the path are adjacent. Moreover, the cliques corresponding to
the endnodes have size 1.
Proof. Consideragraphwithmaximalspectralradiusamongthegraphsonnnodeswithdiameter
D. Since the diameter of the graph is D, there are nodes v0 and vD at distance D. Let Ni be the
set of nodes at distance i from v0, for i = 0,1,...,D. If the graph is not of the claimed form,
then one of the sets Ni contains two nodes that are not adjacent, or there is a node in Ni and a
node in Ni+1 that are not adjacent, or the set ND contains more than one element. In the ﬁrst
two cases this gives a contradiction, since adding the missing edge increases the spectral radius
and leaves the diameter the same. In the last case it also does, since adding edges between all but
one node of ND and all nodes of ND−2 increases the spectral radius, and leaves the diameter the
same. 
LetGnowbeagraphwithmaximalspectralradiusamongallgraphsonnnodeswithdiameter
D. According to Lemma 1 it consists of “pathwise adjacent cliques”. Let us call these cliques
Ni,i= 0,1,...,D, ordered such that all nodes of Ni are adjacent to all nodes of Ni+1, for
i = 0,1,...,D− 1. Let ni be the size of Ni, for i = 0,1,...,D. From Lemma 1 we know that
n0 = nD = 1. The following lemma states that there is only one clique Ni of size bigger than
1.
Lemma 2. There is at most one i such that ni > 1.
Proof. Consider the partition of the nodes into the sets Ni, and the corresponding quotient matrix
of the adjacency matrix of G, i.e., the matrix Q labeled by the sets Ni, where QNi,Nj equals the
average number of neighbours in Nj, of the nodes in Ni. Since here we have that for all i and j
the number of neighbours in Nj is the same for all nodes in Ni (the partition is called regular, or
equitable), it follows that the spectral radius of Q is the same as the spectral radius of G (cf. [4,
p. 79]). Moreover, Q is the following tridiagonal matrix:
Q =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 n1 00 ··· 0
1 n1 − 1 n2 0 ··· 0
0 n1 n2 − 1
...
...
. . .
00 n2
... nD−1 0
. . .
. . .
...
... nD−1 − 11
00 ··· 0 nD−1 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
and if x is a positive eigenvector of Q corresponding to the spectral radius ρ, then the vector
y deﬁned by yv = xi if v is a node in Ni is an eigenvector of the adjacency matrix A of G,
with eigenvalue ρ. Without loss of generality we normalize x in such a way that y has length456 E.R. van Dam / Linear Algebra and its Applications 426 (2007) 454–457
1. Thus ρ = y Ay. Moreover, from the eigenvalue equation Qx = ρx we obtain that ρxi =
ni−1xi−1 + (ni − 1)xi + ni+1xi+1 for i = 1,...,D− 1.
Suppose now that ns > 1 and nt > 1, where s/ = t. Without loss of generality we may assume
that xs  xt. Consider then the graph G  obtained from G by moving one node from clique Nt
to clique Ns. Then G  has the same number of nodes and diameter as G (and it is of similar
form, as in Lemma 1). If A  is the adjacency matrix of G  with spectral radius ρ , then it follows
that if s and t differ by more than one, then ρ   y A y = y Ay + 2xt(ns−1xs−1 + nsxs +
ns+1xs+1 − nt−1xt−1 − (nt − 1)xt − nt+1xt+1) = ρ + 2xt(xs + ρ(xs − xt) )>ρ . Since G has
maximal spectral radius, this gives a contradiction. If s and t differ by one, then we ﬁnd similarly
that ρ   y A y = ρ + 2xt(ρ + 1)(xs − xt)  ρ. This implies that ρ  = ρ, and that y is also an
eigenvector of A  with eigenvalue ρ. It then follows that Ay = A y, which gives a contradiction
by considering the entry of a node in Ns−1 or Ns+1 (depending on whether s is smaller or larger
than t) in these vectors. Thus the made assumption is not true, and the statement is proven. 
From now on, consider graphs of the form in Lemma 1 where at most one of the cliques, say
Na, has size bigger than one. Let m = na be ﬁxed in the remainder, and let b = D − a. Notice
that these are graphs obtained from a complete graph on m + 2 = n − D + 2 nodes by removing
an edge, adding a pendant path of a − 1 edges to one endnode of the removed edge, and adding
a pendant path of b − 1 edges to its other endnode. Thus we have proven the theorem if we can
show that a and b differ by at most one in a graph with maximal spectral radius.
Let Ca,b be the characteristic polynomial of the quotient matrix Q of the above graph (on
n = a + m + b nodes and with diameter D = a + b), i.e., Ca,b(λ) = det(λI − Q). Also, let Pk
be the characteristic polynomial of the path on k nodes.
Lemma 3. If a>b+ 1  1, then Ca,b(λ) − Ca−1,b+1(λ) = (m − 1)(λ + 1)Pa−b−2(λ).
Proof. If b>0 then it follows from expansion of the determinant that Ca,b(λ) = λCa−1,b(λ) −
Ca−2,b(λ) and that Ca−1,b+1(λ) = λCa−1,b(λ) − Ca−1,b−1(λ), and hence it follows that
Ca,b − Ca−1,b+1 = Ca−1,b−1 − Ca−2,b. Thus it follows by induction that Ca,b − Ca−1,b+1 =
Ca−b,0 − Ca−b−1,1.
Now let Tc = Cc,0 − Cc−1,1 for c>0. From expansion of the determinants in Cc,0 and
Cc−1,1 it follows that Tc(λ) = λTc−1(λ) − Tc−2(λ) for c>2. Thus Tc satisﬁes the same re-
currence relation as the characteristic polynomials of the paths. Since T1 = C1,0 − C0,1 = 0
and since it can be shown that T2(λ) = C2,0(λ) − C1,1(λ) = (m − 1)(λ + 1) = (m − 1)(λ +
1)P0(λ),itnowreadilyfollowsthatTc(λ) = (m − 1)(λ + 1)Pc−2(λ).Theclaimedstatementnow
follows. 
To ﬁnish the proof of the theorem, suppose that a and b differ by more than 1 in a graph
with maximal spectral radius. Without loss of generality we may assume that a>b+ 1. Since
the theorem is trivially true for D = n − 1( m = 1), we also assume that m  2. Then the graph
contains a triangle, and hence its spectral radius is at least 2. Since the roots of the characteristic
polynomial of a path are all less than 2, it follows from Lemma 3 that the largest root of Ca−1,b+1
is strictly larger than the largest root of Ca,b, which is a contradiction, and which completes the
proof of the theorem.
Note. After writing this paper, the author was informed about two other papers. Hansen and
Stevanovi´ c[ 6] (see [7] for an extended abstract) obtained the same result in a different manner.
Feng [3] claims the same result, however with an incomplete proof.E.R. van Dam / Linear Algebra and its Applications 426 (2007) 454–457 457
Acknowledgments
The author thanks an anonymous referee for pointing out an error in the original manuscript.
References
[1] D.M. Cvetkovi´ c, M. Doob, H. Sachs, Spectra of Graphs, third ed., Johann Abrosius Barth Verlag, 1995.
[2] E.R. van Dam, R.E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra Appl. 423
(2007) 408–419.
[3] L. Feng, Spectral radius of graphs with given diameter, Ars Combin., in press.
[4] C.D. Godsil, Algebraic Combinatorics, Chapman & Hall, 1993.
[5] J. Guo, J. Shao, On the spectral radius of trees with ﬁxed diameter, Linear Algebra Appl. 413 (2006) 131–147.
[6] P. Hansen, D. Stevanovi´ c, On bags and bugs, Discrete Appl. Math., in press.
[7] P. Hansen, D. Stevanovi´ c, On bags and bugs, Electronic Notes Discrete Math. 19 (2005) 111–116.

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Graphs with given diameter maximizing the spectral radius.

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