Graph catalog: Families of graphs
Version: March 1, 2007
Main Index
Index of graphs
Description of graph entries
Graph operations
Parameter relationships
References
The catalog
This catalog provides information about the minimum rank and other graph
parameters for families of connected graphs. The description of a family
of graphs involves one or more parameters (often the order n of the
graph).
For specific small graphs, please consult the
Graph
catalog: Small graphs.
For general background on the minimum
rank problem, please see [FH]. Proofs of
the results in the catalog can be found in the references
cited in the catalog.
This catalog was developed through the American Institute of Mathematics
workshop, "Spectra of Families of Matrices described by Graphs, Digraphs,
and Sign Patterns," and is hosted by the AIM website. It is edited by
Wayne Barrett, Leslie Hogben, and Hein van der Holst.
The webpage was designed by David Farmer.
Please refer any questions or comments about the content of the catalog,
including corrections or suggestions for additional information, to
Leslie Hogben (lhogben(())iastate.edu).
Please refer any questions or comments about the operation of this website
to David Farmer (farmer(())aimath.org).
AIM workshop participants:
Francesco Barioli, Wayne Barrett, Avi Berman,
Richard Brualdi, Steven Butler,
Sebastian Cioaba, Dragos Cvetkovic, Jane Day,
Louis Deaett, Luz DeAlba, Shaun Fallat, Shmuel Friedland,
Chris Godsil, Jason Grout, Willem Haemers, Leslie Hogben,
InJae Kim, Steve Kirkland, Raphael Loewy, Judith McDonald,
Rana Mikkelson,
Sivaram Narayan, Olga Pryporova, Uri Rothblum, Irene Sciriha,
Bryan Shader, Wasin So, Dragan Stevanovic, Pauline van den Driessche,
Hein van der Holst, Kevin Vander Meulen, Amy Wangsness, Amy Yielding.
Entries in the catalog
A typical entry in this catalog includes the following information
for the graph or family of graphs.
Symbol name 
Common names/symbols are used to describe a graph G in the family.
Sometimes a graph operation is used.

picture 
A picture of the graph, or a representative example from the family.

order  The number of vertices in the graph

mr 
For a real symmetric matrix A, the graph of A,
denoted G(A), is the graph with vertices
{1,...,n } and edges { {i,j }  a_{ij} ≠ 0 and i
≠ j }
(note that the diagonal of A is ignored in determining G(A)).
The minimum rank of G is
mr(G)=min{rank(A) : A ∈ R^{n}, A^{T}=A, and G(A)=G}.

M 
The maximum nullity (= maximum multiplicity of an eigenvalue) of a real symmetric matrix A such that G(A)=G

field independent 
Minimum rank can be defined for symmetric matrices over any field. ``Yes" means the minimum rank of G is the same for all fields.

ξ 
A real symmetric matrix M satisfies the Strong Arnold
Hypothesis provided there does not exist a nonzero symmetric matrix
X satisfying:
 MX = 0,
 M ° X = 0,
 I ° X=0,
where ° denotes the Hadamard (entrywise) product and I is the identity matrix.
The Colin de Verieretype parameter ξ(G) is the maximum multiplicity of 0 as an eigenvalue
among symmetric real matrices that satisfy
 G(A )=G and
 A satisfies the Strong Arnold Hypothesis.

ω 
The largest value of m for which G has a clique of order m (subgraph isomorphic to K_{m}) is called the clique number of G
and denoted by ω(G).

δ 
The minimum degree of a vertex of G is denoted by δ(G).

tw 
A treedecomposition of a graph G=(V,E) is a pair
(T,W), where T is a tree and W={W_{t} : t∈ V(T)} is a family of subsets of V
with the properties:
 ∪ {W_{t} : t∈ V(T)}=V,
 for every edge e∈ E, there is a t∈V(T) such that has both ends of e are in W_{t}, and
 if t_{1},t_{2},t_{3}∈ V(T)
and t_{2} lies on the path from t_{1} to t_{3} in T, then
W_{t1}∩W_{t3}⊆W_{t2}.
The width of a treedecomposition is
max{W_{t}  1 : t∈ V(T)}. The treewidth of G, denoted by tw(G), is the minimum possible width of a treedecomposition of G.

cc 
Clique covering number, or Cliquecover.
A set of subgraphs of G, each of which is a clique and such that every edge of G is
contained in at least one of these cliques, is called a clique covering of G.
The clique covering number of G, denoted by cc(G), is the smallest number of
cliques in a clique covering of G.

diam 
The distance between two vertices in a graph G is the
number of edges in a shortest path between them.
The diameter of G, diam(G), is the maximum distance between any two vertices
of G.

maxinducedpath 
The maximum number of edges in a path that is an induced subgraph of G.

Z 
Colorchange rule:
If G is a graph with each vertex colored either white or black, u is a black vertex of G, and exactly one neighbor v of u is white, then change the color of v to black.
Given a coloring of G, the derived coloring is the result of applying the colorchange rule until no more changes result.
A zero forcing set for a graph G is a subset of vertices Z such that if initially the vertices in Z are colored black and the remaining vertices are colored white, the derived coloring of G is all black.
Z(G) is the minimum of Z over all zero forcing sets Z⊆ V(G).

α 
An induced subgraph H of a graph G
is a coclique or independent set of vertices
if H has no edges.
The largest value of m for which a coclique with m
vertices exists is called the vertex independence number
of G and denoted by α(G).


Notes: 
Any other comments.

Graph operations
The following graph operations are used to construct families:
 The complement of a graph G=(V,E) is the
graph G
= (V,E),
where E
consists of all two element sets from V that are not in E.
 The line graph of a graph G=(V,E),
denoted L(G), is the graph having vertex set E,
with two vertices in L(G) adjacent if and only if the
corresponding edges are adjacent in G.
 The Cartesian product of two graphs G and H,
denoted G ☐ H,
is the graph with vertex set V(G) × V(H) such that
(u,v) is adjacent to (u',v') if and only if
(1) u=u' and vv' ∈ E(H),
or
(2) v=v' and uu' ∈ E(G).
 The strong product of two graphs G and H, denoted
G ⊠ H, is the graph with vertex set V(G) × V(H) such
that (u,v) is adjacent to (u',v') if and only if
(1) uu' ∈ E(G)
and vv' ∈ E(H),
or
(2) u=u' and vv' ∈ E(H),
or
(3) v=v'
and uu' ∈ E(G).
 The corona of G with H, denoted
G ° H, is the graph of order GH + G
obtained by taking one copy of G and G copies
of H, and joining all the vertices in the ith
copy of H to the ith vertex of G.
Parameter Relationships
The following relationships between the parameters are known:
 mr(G) + M(G) = G (where G denotes the order of G)
 M(G) ≤ Z(G) [AIM]
 ξ(G) ≤ M(G) [BFH3]
 ω(G) 1 ≤ ξ(G) [BFH3]
 δ(G) ≤ tw(G)
 mr(G) ≤ cc(G)
 diam(G) ≤ maxinducedpath(G) ≤ mr(G)
 ξ(G) ≤ G  α(G) + 1 [BFH3]
References
[AIM] AIM Minimum rank  special graphs work group. Zero forcing sets and the minimum rank of graphs. Preprint.
[BFH] F. Barioli, S. Fallat, and L. Hogben.
Computation of minimal rank and path cover number for graphs.
Linear Algebra and Its Applications,
392: 289303, 2004.
[BFH2] F. Barioli, S. Fallat, and L. Hogben.
On the difference between the maximum multiplicity and path cover number for
treelike graphs.
Linear Algebra and Its Applications 409: 1331, 2005.
[BFH3] F. Barioli, S. Fallat, and L. Hogben.
A variant on the graph parameters of Colin de Verdi`ere:
Implications to the minimum rank of graphs.
Electronic Journal of Linear Algebra,
13: 387404, 2005.
[BHL] W. Barrett, H. van der Holst and R. Loewy.
Graphs whose minimal rank is two.
Electronic Journal of Linear Algebra,
11: 258280, 2004.
[FH] S. Fallat and L. Hogben.
The Minimum Rank of Symmetric Matrices Described by a Graph: A Survey. Preprint.
[H] L. Hogben.
Spectral graph theory and the inverse eigenvalue problem of a
graph.
Electronic Journal of Linear Algebra, 14:1231, 2005.
[HvdH] L. Hogben and H. van der Holst.
Forbidden minors for the class of graphs G with
ξ(G) ≤ 2. To appear in Linear Algebra and Its Applications.
[JLD] C. R. Johnson and A. Leal Duarte.
The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree.
Linear and Multilinear Algebra 46: 139144, 1999.
[JS] C. R. Johnson and C. M. Saiago.
Estimation of the maximum multiplicity of an eigenvalue in terms of the vertex degrees
of the graph of the matrix.
Electronic Journal of Linear Algebra,
9:2731, 2002.
path on
vertices

Order : 
 mr : 
 M :  1  field independent :  yes  ξ :  1  ω :  2  δ :  1  tw :  1  cc : 
 Diameter : 
 maxinducedpath : 
 Z :  1  α : 
 Notes : 
is displayed.

 
cycle

Order : 
 mr : 
 M :  2  field independent :  yes  ξ :  2  ω :  2  δ :  2  tw :  2  cc : 
 Diameter : 
 maxinducedpath : 
 Z :  2  α : 
 Notes : 
is displayed.
.

 
complete graph on
vertices

Order : 
 mr :  1  M : 
 field independent :  yes  ξ : 
 ω : 
 δ : 
 tw : 
 cc :  1  Diameter :  1  maxinducedpath :  1  Z : 
 α :  1  Notes : 
is displayed.

 
complete bipartite graph on
vertices

Order : 
 mr :  2  M : 
 field independent :  yes  ξ : 
(for
)
 ω :  2  δ : 
 cc : 
 Diameter :  2  maxinducedpath :  2  Z : 
 α : 
 Notes : 
is displayed.
.

 
Complete multipartite graph 
Order : 
 mr : 
 field independent :  no  δ : 
 Notes : 
where
.
is shown.
.
.
References:[BvdHL04], [FH]

 
th hypercube

Order : 
 mr : 
 M : 
 ω :  2  δ : 
 cc : 
(= # edges)
 Diameter : 
 Z : 
 α : 
 Notes : 
is displayed.
.
Reference:[AIM]

 
th supertriangle

Order : 
 mr : 
 M : 
 ξ : 
 ω :  3  δ :  2  cc : 
 Diameter : 
 Z : 
 Notes : 
is shown.
minimum rank is same for all infinite fields.
Reference: [AIM]

 
wheel on
vertices

Order : 
 mr : 
 M :  3  field independent :  no (see notes)  ξ :  3  ω :  3  δ :  3  tw :  3  cc : 
 Diameter :  2  maxinducedpath : 
 Z :  3  α : 
 Notes : 
is displayed.
.
mr
mr
.
References: [H]; [BFH05a, Thm 3.16] for
.

 
pineapple (
)

Order : 
 mr : 
 M : 
 field independent :  yes  ω : 
 δ :  1  cc : 
 Diameter :  2  maxinducedpath :  2  Z : 
 α : 
 Notes :  The
pineapple is
where the intersection is the single vertex
.
is shown.
can be covered by
and
, so mr
.
All vertices of
except
and one other, together with all but one of the
vertices of degree 1, is a zero forcing set.

 
Moebius ladder 
Order : 
 mr : 
 M :  4  ξ :  4  ω :  2  δ :  3  cc : 
(= # edges)
 Z :  4  Notes :  5th Möbius ladder is shown.
Reference: [AIM]

 
tree 
mr : 
 M : 
 field independent :  yes  ξ :  2 (for
not a path)
 ω :  2  δ :  1  tw :  1  cc : 
 Z : 
 Notes : 
: there is a set of vertices whose deletion leaves paths}
= path cover number.
References: [JLD];
for algorithms for computation of
, see [JS] or [FH]; for
, see [AIM]; for
see [BFH3].

 
linear 2tree 
mr : 
 M :  2  field independent :  yes  ξ :  2  δ :  2  Z :  2  Notes :  A linear
tree is a
connected graph
that can be embedded
in the plane such that the graph obtained from the dual of
after
deleting the
vertex corresponding to the infinite face is a path.
Reference: [HvdH07]

 
clawfree blockclique (1chordal) graph 
mr :  = cliquecover(
)
 M : 
 field independent :  yes  cc :  # of blocks  Z : 
 Notes :  For the blockclique graph
shown,
mr(
)=4.
This result requires clawfree or equivalently no vertex in more than 2 blocks.
A clawfree blockclique is the line graph of a tree.
reference: [AIM]

 
complement of
cycle

Order : 
 mr :  3  M : 
 field independent :  no (see notes)  ω : 
 δ : 
 Diameter :  2  maxinducedpath :  3  Z : 
 α :  2  Notes : 

 
complement of a tree 
mr : 
if
 M : 
 maxinducedpath :  3 if
 Z : 
 α :  2  Notes : 
.
Reference: [AIM]

 
line graph of complete graph 
Order : 
 mr : 
 M : 
 δ : 
 Diameter :  2  Z : 
 Notes : 
is shown
Reference: [AIM]

 
line graph of a tree 
Order : 
 mr : 
 number of pendent vertices of
 M :  number of pendent vertices of
 1
 field independent :  yes  ω :  maximum degree of vertex of
 cc : 
 number of pendent vertices of
 Z :  number of pendent vertices of
 1
 Notes :  Reference: [AIM]

 
Cartesian product of two paths 
Order : 
 mr : 
 M : 
 ω :  2  δ :  2  cc :  = # of edges  Diameter : 
 Z : 
 Notes : 
is shown
Reference: [AIM]

 
Cartesian product of a complete graph and a path 
Order : 
 mr : 
 M : 
 field independent :  no (see notes)  ξ : 
 ω : 
 δ : 
 cc : 
 Diameter : 
 maxinducedpath : 
 Z : 
 α :  2  Notes : 
is displayed.
Minimum rank over
is different
References: [AIM]

 
Cartesian product of a cycle and a path 
Order : 
 mr : 
 M : 
 ω :  2 (
)
 δ :  3  cc :  = # of edges (if
)
 Diameter : 
 Z : 
 Notes : 
is shown.
Reference: [AIM]

 
cartesian product of two complete graphs 
Order : 
 mr : 
 M : 
 ω : 
 δ : 
 Z : 
 Notes : 
is shown.
Reference: [AIM]

 
Cartesian product of a cycle and a complete graph 
Order : 
 mr : 
 M : 
 ω : 
 δ : 
 Z : 
 Notes : 
is shown.
Reference: [AIM]

 
strong product of two paths 
Order : 
 mr : 
 M : 
 ω :  4  δ :  3  cc : 
 Diameter : 
 Z : 
 Notes : 
is shown.
minimum rank same for all infinite fields
Reference: [AIM]

 
corona of
by
(
)

Order : 
 mr : 
 M : 
 field independent :  yes  ω : 
 δ : 
 cc : 
 Diameter :  3  maxinducedpath :  3  Z : 
 α : 
 Notes : 
is shown
References: [AIM]

 
corona of
with
(
sun)

Order : 
 mr : 
(for
)
 M : 
(for
)
 field independent :  yes  ω :  2 (for
)
 δ :  1  tw :  2  cc : 
(for
)
 Diameter : 
 maxinducedpath : 
 α : 
 Notes : 
is displayed.
mr(
)=4.
References: [BFH04] and [BFH05]

 
necklace 
Order : 
 mr : 
 M : 
 ω :  3  δ :  3  cc : 
 maxinducedpath : 
 Z : 
 Notes :  The necklace with
diamonds is a 3regular graph that can be constructed from a
cycle by appending
extra vertices. Each "extra" vertex is adjacent to 3 sequential cycle vertices. The adjacency matrix + I achieve minimum rank.

 