Spectral Graph Theory and the
Inverse Eigenvalue Problem of a Graph
Leslie Hogben
Department of Mathematics,
Iowa State University, Ames, IA 50011
LHogben@iastate.edu
http://www.math.iastate.edu/lhogben/research/
Directions in Combinatorial Matrix Theory
Banff International Research Station
May 8, 2004
Definitions and Notation
A graph G = (V, E) means a simple undirected graph (no loops, no
multi-edges), vertices V  Z+. The order is the number of vertices.
The degree of vertex k, deg k, is the number of edges incident
with k.
A tree T is a connected graph with no cycles.
Pn is a path on n vertices
Cn is a cycle on n vertices
Kn is the complete graph on n vertices.
K1,n is a star on n+1 vertices
Wn is a wheel on n vertices
G – v is the graph resulting from the deletion of vertex v and all its
incident edges from G. The induced subgraph <S> is the result of
deleting all vertices not S from G.
G is regular of degree r if every vertex has degree r.
Unless otherwise noted, a graph is assumed connected, because
each connected component can be analyzed separately.
Spectral Graph Theory
Spectral graph theory uses the spectra of matrices associated with
the graph, such as the adjacency matrix, the Laplacian matrix, or
the normalized Laplacian, to provide information about the graph.
One goal is to characterize a graph or obtain information about the
graph from the spectra of these matrices. There also important
applications to other fields such as chemistry.
Hundreds of papers and several books such as Spectra of Graphs
by Cvetkovic, Doob, and Sachs, Recent Results in the Theory of
Graph Spectra by Cvetkovic, Doob, Gurman, Torgasev, and
Spectral Graph Theory by Chung have been written on spectral
graph theory.
Let G be a graph with vertices {1,…,n}.
Matrices associated with G:
Let A denote the adjacency matrix of G ( aij = 1 if {i,j} is an edge,
else 0). Denote its ordered spectrum
 (A)  {1,...n } with k  k1
Let D denote the diagonal matrix whose ith entry is the degree of
vertex i. The normalized adjacency matrix is Aˆ  D 1 A D 1
with ordered spectrum  ( Aˆ )  {
ˆ1,...
ˆ n } with 
ˆ k  
ˆ k1
Let A = D + A. Denote its ordered spectrum  (A)  { ,... }
1
n
The normalized form of A is
1
1
1
1
ˆ
A  D A D  I  D A D  I  Aˆ
ˆ1 ,...
ˆ n } with 
ˆ k  
ˆ k1
 (Aˆ )  {
D1 is an eigenvector for eigenvalue 1 of Aˆ (and 2 of Aˆ ).
(1 denotes the vector all of whose entries are 1.)
Example

 0

 1
2 3
 1
ˆ
A  
2 3
 1
2 3
 1

2 3
0

1
A  1

1

1
1 1 1 1

0 1 0 1
1 0 1 0

0 1 0 1

1 0 1 0
1 
  (A)  {2,1 5,0,0,1 5}
2 3 2 3 2 3 2 3 
1
1 
0
0
3
3 
4 1 1 1 1



1
1
0
0 
1 3 1 0 1

3
3
 A  1 1 3 1 0
1
1 


0
0
1 0 1 3 1
3
3 



1
1
1
1
0
1
3


0
0 

3
3
9  17
9  17
2
1
 (A )  {1,
, 3, 3,
}
 ( Aˆ )  { , ,0, 0,1}
2
2
3 3
1
1
1
The following graph parameters are determined by the adjacency
matrix A and its characteristic polynomial
p(x)  x n  bn1x n1  bn2 x n2  ... b1 x  b0
the number of edges = - bn2
tr A 2   i 2
=

2
2
bn3
tr A 3   i 3
the number of triangles = =

2
6
6
The first equality is obtained by viewing the coefficient of x k as
the sum of the principal minors of order k, and the second is
obtained by considering walks.
Unfortunately these results do not extend cleanly to longer cycles,
as can be seen by considering the 4-cycle. However, counting
disjoint cycles can be used to evaluate the coefficients.
If two graphs have different spectra (equivalently, different
characteristic polynomials) then they are not isomorphic.
However, non-isomorphic graphs can be cospectral:
p(x)  x 6  7x 4  4 x 3  7x 2  4x 1
Examples of spectrally determined graphs:
Complete graphs
Empty graphs
Graphs with one edge
Graphs missing only 1 edge
Regular of degree 2
Regular of degree n-3
m Kn
Kn,n,…,n
However, “most” trees are not spectrally determined.
(M) denotes the spectral radius of M.
Perron-Frobenius Theorem
Let M be an irreducible non-negative n  n matrix. Then
a) (M) > 0
b) (M) is an eigenvalue of M
c) (M) is algebraically simple as an eigenvalue of M
d) there is a positive vector x such that M x = (M) x
The matrices A, Aˆ , A, Aˆ are all nonnegative
(and irreducible if G is connected).
The Perron root of M is the largest eigenvalue of M (for M = A it
is called the index of G).
Additional matrices associated with G:
L = D – A is the Laplacian of G
The normalized Laplacian of G is
1
1
1
1
ˆ
L  D L D I D A D
For any graph Lˆ  Aˆ  2I and Lˆ  Aˆ  I so
ˆ = 2 - 
ˆ nk1 =1- 
ˆ nk1

k
If G is regular of degree r then
A  rI  A, so  k  r   k
L  rI  A, so  k  r   k
1
1
1
1
Aˆ  A, so 
ˆ k   k Aˆ  A, so 
ˆ k  k
r
r
r
r
1
1
ˆ
ˆ
L  L , so k   k
r
r
Example

1
1
1
1 




 1

2
3
2
3
2
3
2
3


1
1
 1
1

0
 
 2 3
3
3 
 1

1
1
ˆ
L  

1

0 
3
3
 2 3

1
1
 1
0

1
 
 2 3
3
3 
 1

1
1

0

1 

 2 3

3
3
4 5
4 1 1 1 1
 ( Lˆ )  {0,1,1, , }
3 3


1 3 1 0 1
L  1 1 3 1 0   (L )  {0, 3, 3, 5,5}


1 0 1 3 1


1
1
0
1
3


The matrices A, D , A , Aˆ , L, Lˆ
are also connected via the incidence matrix.
The (vertex-edge) incidence matrix N of graph G with n vertices
and m edges is the n  m 0,1-matrix with rows indexed by the
vertices of G and columns indexed by the edges of G such that the
v,e entry of N is 1 (respectively, 0) if edge e is (respectively, is not)
incident with vertex v. Then
1
1
1
1 T
ˆ
N NT = D + A = A.
A  D A D  ( D N)( D N)
If G’ is any orientation of G and N’ is the oriented incidence matrix
then
N’ N’ T = D – A = L, and
L=
1
1
1
1
T
D
L
D

(
D
N')(
D
N'
)
So
are all positive semidefinite, and so have
A, Aˆ , L,eigenvalues.
Lˆ
nonnegative
The following facts are straightforward (if G is connected and
not Kn).
 (Aˆ )  [0,2] and 
ˆ n  2, so  ( Aˆ )  [1,1] and 
ˆn  1
ˆ  0 and   0
 ( Lˆ )  [0, 2] and 
1
1
ˆ n
 
i
ˆ 1
0  
2
ˆ  n and 
ˆ  2 if and only if G is bipartite
2  
n
n
n 1
If G is not connected, the multiplicity of 0 as an eigenvalue of L
is the number of connected components. For each of the
matrices, the spectrum of is the union of the spectra of the
components.
The second smallest eigenvalue of L, 2 , is called the algebraic
connectivity. The vertex connectivity, 0 , is the minimum
number of vertices in a cutset (for a graph that is not the complete
graph).
Theorem If G is not Kn, the vertex connectivity is greater than or
equal to the algebraic connectivity, i.e., 2   0
Example
 (L (W5 ))  {0, 3, 3, 5,5}
2 (W5 )  3   0 (W5 )
The distance between two vertices in a graph is the length of the
shortest path between them. The diameter of a graph G,
diam(G), is maximum distance between any two vertices of G.
Theorem The diameter of G is less than the number of distinct
eigenvalues of the adjacency matrix of G.
There are also several other diameter results involving the
Laplacian and normalized Laplacian.
Inverse Eigenvalue Problem of a Graph (IEPG)
Sn = the set of symmetric real n  n matrices
For B  Sn, the graph of B, G(B), is the graph with vertices
{1,…,n} and edges E = { {i,j} | aij  0}.
0
1
0
0


1 3.1 1.5 2
B  
0 1.5
1
1



2
1
0
0

1
G(B) =
For G a graph with vertices {1,…,n}, the set of symmetric
matrices of the graph is
S(G) = { B  Sn | G(B) = G}.
The Inverse Eigenvalue Problem of a Graph is to characterize
the possible spectra of matrices in S(G).
Note that A,A, Aˆ , L, Lˆ are all in S(G).
L is a generalized Laplacian matrix of G if L has nonpositive offdiagonal elements and L  S(G). Note that in this case, -L has
non-negative off-diagonal elements and there is a real number c
such that cI - L is non-negative, so if G is connected, the least
eigenvalue of L is simple.
Theorem Let L be a generalized Laplacian matrix of the graph G.
If G is 3-connected and planar then  (L) has multiplicity less
2
than 4.
Much recent work with generalized Laplacians is based on Colin
de Verdière matrices.
There interesting connections between the work on generalized
Laplacians and the Inverse Eigenvalue Problem of a Graph.
Definitions and Notation
Let B  Sn .
 (B) = {1, …, n} is the ordered spectrum of B (k < k+1 ).
mB() = the multiplicity of  as an eigenvalue of B
The eigenvalue  is simple if mB() = 1.
M(G) = max{mB() | B  S(G) } maximum multiplicity of G
mr(G) = min{ rank B | B  S(G) } minimum rank of G
M(G) + mr(G) = n.
If H is an induced subgraph of G then mr(H) < mr(G).
mr(Kn) = 1 and mr(Pn) = n - 1.
Theorem [Fiedler]
If mB() = 1 for all   (B) for all B  S(G), then G = Pn.
Equivalently, mr(G) = n - 1 implies G = Pn .
Theorem [Barrett and Loewy]
mr(G) = 2 if and only if G is not Kn and does not contain as an
induced subgraph any of:
P4, K3,3,3 = complete tripartite graph, or
Most of the progress on the Inverse Eigenvalue Problem of a
Graph is for trees.
Recall that for any graph, the diameter of G is less than the number
of distinct eigenvalues of A, and the proof extends to show
diam(G) < the number of distinct eigenvalues of any non-negative
matrix B  S(G).
If T is a tree and B  S(T), it is possible to find a 1,-1 diagonal
matrix S with S 1BS non-negative.
Theorem [Johnson and Leal Duarte] If T is a tree, for any
B  S(T), the diameter of T is less than the number of distinct
eigenvalues of B.
Thus, diam(T) < mr(T)
If B is an n  n matrix, B(k) is the (n -1) (n -1) matrix obtained
from B by deleting row and column k. If B  S(G), then
B(k)  S(G-k).
Interlacing Theorem
Let B  Sn, k  {1,…,n}.
If the eigenvalues of B are 1 < 2 < … < n and
the eigenvalues of B(k) are 1 < 2 < … < n-1, then
1 < 1 < 2 < 2 < 3 < … < n-1 < n.
Corollary
mB(k)()  {mB() -1, mB(), mB() +1}.
k is a Parter-Wiener (PW) vertex of B for eigenvalue 
if mB(k)() = mB() + 1.
k is a strong PW vertex of B for  if k is a PW vertex of B for 
and  is an eigenvalue of at least three components of B(k).
Parter-Wiener Theorem
If T is a tree, B  S(T) and mB() > 2, then there is a strong PW
vertex of B for .
Corollary
If T is a tree, B  S(T), and (B) = (1, …, n) with k < k+1
then 1 and n are simple eigenvalues
H(G) = {k  V(G) | deg k > 3} is called the set of high degree
vertices of G. Only high degree vertices can be strong PW
vertices.
Example
The star on n+1 vertices K1,n has only one high degree vertex,
say 1, so this vertex must be PW for any multiple eigenvalue of B
with G(B) = K1,n. By choosing the diagonal elements of B for
2,…,n to be 0, we obtain mB(k)(0) = n and so mB(0) = n -1 and
mr(K1,n) = 2. (In this case the other two eigenvalues are
necessarily simple).
The Parter-Wiener Theorem need not be true for graphs that are
not trees.
Example
For A the adjacency matrix of C4, mA(0) = 2 but there is no PW
vertex since C4 - k is P3 for any vertex k.
P(G), the path cover number of G, is the minimum number of
vertex disjoint paths occurring as induced subgraphs of
G that cover all the vertices of G.
(G) = max{p-q | there is a set of q vertices whose deletion
leaves p paths}
Theorem [Johnson and Leal Duarte]
Let T be a tree. Then M(T) = P(T) = (T).
For any graph G, (G) < MG).
[Barioli, Fallat, Hogben] For any graph G, (G) < PG).
Theorem [Johnson, Leal Duarte, Saiago]
The possible ordered multiplicity lists of the following families
of trees have been determined. Each possible list can be realized
for any list of real numbers in order.
Paths
Stars
Double Paths
Generalized Stars
Double Generalized Stars
However, there exist trees for which an ordered multiplicity list is
possible but not attainable for all such ordered real number lists.
Example [Barioli and Fallat]
0 0 1 0 0 1 0

0 0 1 0 0 0 0
1 1 0 1 0 0 0

0 0 1 0 0 0 0

0 0 0 0 0 1 0

A
1 0 0 0 1 0 1

0 0 0 0 0 1 0

0 0 0 0 0 0 0
1 0 0 0 0 0 0

0 0 0 0 0 0 0
0 1 0

0 0 0
0 0 0

0 0 0

0 0 0
0 0 0

0 0 0

0 1 0
1 0 1

0 1 0
A  S(BF).
 (A)  { 5, 2, 2, 0,0, 0,0, 2, 2, 5}
so A has ordered multiplicity list 1, 2, 4, 2, 1
but if B  S(BF) has the five distinct eigenvalues
1  2  3  4  5 and
m B (1 )  1
m B (2 )  2
m B (3 )  4
m B (4 )  2
m B (5 )  1
then 1  5  2  4
We now examine matrices realizing minimum rank and having
some special form such as A, a 0,1 matrix in S(G), or a
generalized Laplacian of G. Note that finding a matrix in S(G)
with non-negative off-diagonal elements and minimum rank is
equivalent to finding a generalized Laplacian of minimum rank.
Theorem [Hogben] If T is a tree and A is its adjacency matrix then
there exists a 0,1 diagonal matrix D such that m0(A+ D) = M(T),
and thus rank (A+D) = mr(T).
Theorem If T is a tree and A is its adjacency matrix then there
exists a 0,1 diagonal matrix D such that m0(A+ D) = M(T), and
thus rank (A+D) = mr(T).
Proof: There exists a set Q of q vertices such that T – Q consists of
p disjoint paths and p – q = M(T). For each path, remove alternate
interior vertices so that the result is isolated vertices (and one path
of 2 vertices if the path had an even number of vertices originally).
Let Q’ be the set of q’ vertices consisting of the original q vertices
and the additional alternate interior vertices deleted. Then Q’ has
the property that T – Q consists of p’ disjoint paths of 1 or 2
vertices and p’ – q’ = M(T). Chose the diagonal of D to be 0 for
isolated vertices and 1 for vertices in a path of 2 vertices in T – Q’.
Then 0 is an eigenvalue of each of the p’ paths and, by interlacing,
m0(A+D) = M(T). rank (A+D) = n – m0(A+D) = n – M(T) = mr(T).
By using the algorithm of Johnson and Saiago for producing the
set Q of vertices to delete to obtain , we obtain the following
algorithm for producing a 0,1 matrix M in S(T).
Let T(v) = degT(v) - degH(T)(v).
To start, set T’ = T and Q = .
Repeat :
1) Q’ = {v | T’ (v) > 2}
2) Set Q = Q  Q’
3) Set T’ = T’ - Q’
Until Q’ = .
In each path, remove alternate interior vertices and add these to Q.
D = diagonal(d1,…,dn) where
1
d k  
0
k  Q and in T  Q, k is in a path of 2 vertices
otherwise
Example The shaded vertices are in Q and the red vertices have
diagonal entry assigned 1 (all other diagonal entries are 0).
Removed 1st iteration
Removed 2nd iteration
Removed alternate interior
Diagonal entry assigned 1
In fact, it is not true for all graphs G that there is a diagonal
matrix D with with rank(A+D) = mr(G).
Example
H=
d1 1

1 d2
1 1
rank 
1 0

1 0

1 1
1
1
1
1
0
0
d3
1
1
1
d4
1
1
1
d5
0
1
1
1 

1 
0 
 3
1 

1 
d6 

However, there is a matrix M  S(H) with
rank(M) = 2 = mr(H) and all off-diagonal entries nonnegative (so a scalar translation of M is non-negative).
And L = -M is a generalized Laplacian of minimum rank.
15 1 3

1 1 1
3 1 0
M  
4 0 1

4 0 1

3 1 0
4
4
0
0
1
1
1
1
1
1
1
1
3

1
0

1

1
0

 (M )  {8  6 3,0, 0,0,0, 8  6 3}
H=
Question:
For every graph G:
Is there a matrix M  S(G) with
rank(M) = mr(G) and all off-diagonal entries non-negative?
Equivalently, is there a generalized Laplacian matrix L of G
with rank(L) = mr(G) ?