Spectra of general hypergraphs

0.16 MB·

by  Anirban Banerjee

Spectra of general hypergraphs
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2 Anirban Banerjee1, 2, Arnab Char1, and Bibhash Mondal1

r
p
A 1Department of Mathematics and Statistics
2Department of Biological Sciences
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Indian Institute of Science Education and Research Kolkata

] Mohanpur-741246, India
P
S [email protected], [email protected],
.
h
[email protected]
t
a
m
[ April 27, 2016

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v
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3 Abstract
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2 Here, we show a method to reconstruct connectivity hypermatri-
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ces of a general hypergraph (without any self loop or multiple edge)
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1 using tensor. We also study the different spectral properties of these
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hypermatrices and find that these properties are similar for graphs
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1 and uniform hypergraphs. The representation of a connectivity hy-
:
v permatrix that is proposed here can be very useful for the further
i development in spectral hypergraph theory.
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r
a AMS classification: 05C65, 15A18
Keywords: Hypergraph, Adjacency hypermatrix, Spectral theory of hyper-
graph, Laplacian Hypermatrix, normalised Laplacian
1 Introduction
Spectral graph theory has a long history behind its development. In spectral
graph theory, we analyse the eigenvalues of a connectivity matrix which is
uniquely defined on a graph. Many researchers have had a great interest to
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study the eigenvalues of different connectivity matrices, such as, adjacency
matrix, Laplacian matrix, signless Laplacian matrix, normalised Laplacian
matrix, etc. Now, a recent trend has been developed to explore spectral
hypergraph theory. Unlike in a graph, an edge of a hypergraph can be con-
structed with more than two vertices, i.e., the edge set of a hypergraph is the
subset of the power set of the vertex set of that hypergraph [20]. Now, one of
the main challenges is to uniquely represent a hypergraph by a connectivity
hypermatrix or by a tensor, and vice versa. It is not trivial for a non-uniform
hypergraph, where the cardinalities of the edges are not the same. Recently,
the study of the spectrum of uniform hypergraphs becomes popular. In a
(m-) uniform hypergraph, each edge contains the same, (m), number of ver-
tices. Thus a m-uniform hypergraph of order n can be easily represented by
a m order n dimensional connectivity hypermatrix (or tensor). In [7], the
results on the spectrum of adjacency matrix of a graph are extended for uni-
form hypergraphs by using characteristic polynomial. Spectral properties of
adjacency uniform hypermatrix are deduced from matroids in [14]. In 1993,
FanChung defined Laplacian of a uniform hypergraph by considering various
homological aspects of hypergraphs and studied the eigenvalues of the same
[5]. In [9, 10, 16, 17, 11], different spectral properties of Laplacian and sign-
less Laplacian of a uniform hypergraph, defined by using tensor, have been
studied. In 2015, Hu and Qi introduced the normalised Laplacian of a uni-
form hypergraph and analysed its spectral properties [8]. The important tool
that has been used in spectral hypergraph theory is tensor. In 2005, Liqun
Qi introduced the different eigenvalues of a real supersymmetric tensor [15].
The various properties of the eigenvalues of a tensor have been studied in
[3, 13, 4, 21, 22, 12, 18, 19].
But, still the challenge remains to come up with a mathematical frame-
work to construct a connectivity hypermatrix for a non-uniform hypergraph,
such that, based on this connectivity hypermatrix the spectral graph theory
for a general hypergraph can be developed. Here, we propose a unique rep-
resentation of a general hypergraph (without any self loop or multiple edge)
by connectivity hypermatrices, such as, adjacency hypermatrix, Laplacian
hypermatrix, signless Laplacian hypermatrix, normalised Laplacian hyper-
matrix and analyse the different spectral properties of these matrices. These
properties are very similar with the same for graphs and uniform hyper-
graphs. Studying thespectrum ofauniformhypergraphs couldbeconsidered
as a special case of the spectral graph theory of general hypergraphs.
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2 Preliminary
Let R be the set of real numbers. We consider an m order n dimensional
hypermatrix A having nm elements from R, where
A = (a ),a ∈ R and 1 ≤ i ,i ,...,i ≤ n.
i1,i2,...,im i1,i2,...,im 1 2 m
Let x = (x ,x ,...,x ) ∈ Rn. If we write xm as a m order n dimension
1 2 n
hypermatrix with i ,i ,...,i entry x ,x ,...,x , then Axm 1, where the
1 2 m i1 i2 im −
multiplication is taken as tensor contraction over all indices, is a n tuple
whose i-th component is
n
a x x ...x .
ii2i3...im i2 i3 im
i2,i3X,...,im=1
Definition 2.1. Let A be a nonzero hypermatrix. A pair (λ,x) ∈ C×(Cn\
{0}) is called eigenvalue and eigenvector (or simply an eigenpair) if they
satisfy the following equation
Axm 1 = λx[m 1].
− −
Here, x[m] is a vector with i-th entry xm. We call (λ,x) is a H-eigenpair (i.e.,
i
λ and x are called H-eigenvalue and H-eigenvector, respectively) if they are
both real.
Definition 2.2. Let A be a nonzero hypermatrix. A pair (λ,x) ∈ C×(Cn\
{0}) is called an E-eigenpair (where λ and x are called E-eigenvalue and
E-eigenvector, respectively) if they satisfy the following equations
Axm 1 = λx,
−
n
x2 = 1.
i
Xi=1
We call (λ,x) a Z-eigenpair if both of them are real.
From the above definitions it is clear that, a constant multiplication of an
eigenvector is also an eigenvector corresponding to a H-eigenvalue, but, this
is not always true for E-eigenvalue and Z-eigenvalue. Now, we recall some
results that are used in the next section.
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Theorem 2.1 ([15]). The eigenvalues of A lie in the union of n disks in C.
These n disks have the diagonal elements of the supersymmetric tensor as
their centres, and the sums of the absolute values of the off-diagonal elements
as their radii.
The above theorem helps us to bound the eigenvalues of a tensor.
Lemma 2.1. Let A be an m order and n dimensional tensor and D =
diag(d ,...,d ) be a positive diagonal matrix. Define a new tensor
1 n
m 1
−
B = A.D (m 1).D...D
− −
z }| {
with the entries
B = A .
i1i2...im i1i2...imdi−1(m−1)di2...dim
Then A and B have the same H-eigenvalues.
Proof. From the remarks of lemma 3.2 in [21].
Some results of spectral graph theory1 are also hold for general hyper-
graphs. If λ is any eigenvalue of an adjacency matrix of a graph G with the
maximal degree ∆ then λ 6 ∆. For a k-regular graph k is the maximum
eigenvalue with a constant eigenvector of the adjacency matrix of that graph.
If λ and µ are the eigenvalues of the adjacency matrices represent the graphs
G and H, respectively, then λ+µ is also an eigenvalue of the same for G(cid:3)H,
the cartesian product of G and H. All the eigenvalues of a Laplacian matrix
ofagrapharenonnegative andavery roughupperboundofthese eigenvalues
is 2∆, whereas, the eigenvalue of a normalised Laplacian matrix of a graph
lies in the interval [0,2]. Zero is always an eigenvalue for both, Laplacian
and normalised Laplacian matrices, of a graph, with a constant eigenvector.
If A and L are the normalised adjacency matrix and normalised Laplacian
matrix, respectively, of a graph (such that L = 1 − A) then the spectrum
of A, σ(A) = 1−σ(L). If M be any connectivity matrix of a graph with r
connected components then σ(M) = σ(M )∪σ(M )···∪σ(M ), where M
1 2 r i
is the same connectivity matrix corresponding to the component i.
1For different spectral properties of a graph see [2, 6]
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3 Spectral properties of general hypergraphs
Definition 3.1. A (general) hypergraph G is a pair G = (V,E) where V
is a set of elements called vertices, and E is a set of non-empty subsets of
V called edges. Therefore, E is a subset of P(V) \ {∅}, where P(V) is the
power set of V.
Example 3.1. Let G = (V,E), whereV = {1,2,3,4,5}andE = {1},{2,3},{1,4,5} .
Here, G is a hypergraph of 5 vertices and 3 edges. (cid:8) (cid:9)
3.1 Adjacency hypermatrix and eigenvalues
Definition 3.2. Let G = (V,E) be a hypergraph where V = {v ,v ,...,v }
1 2 n
and E = {e ,e ,...,e }. Let m = max{|e | : e ∈ E} be the maximum
1 2 k i i
cardinality of edges, m.c.e(G), of G. Define the adjacency hypermatrix of G
as
A = (a ), 1 ≤ i ,i ,...,i ≤ n.
G i1i2...im 1 2 m
For all edges e = {v ,v ,...,v } ∈ E of cardinality s ≤ m,
l1 l2 ls
s m!
a = , where α = ,
p1p2...pm α k !k !...k !
X 1 2 s
k1,k2,...,ks 1,Pki=m
≥
and p ,p ,...,p are chosen in all possible way from {l ,l ,...,l } with at
1 2 m 1 2 s
least once for each element of the set. The other positions of the hypermatrix
are zero.
Example 3.2. Let G = (V,E) be a hypergraph in example 3.1. Here, the
maximum cardinality of edges is 3. The adjacency hypermatrix of G is A =
G
(a ), where 1 ≤ i ,i ,i ≤ 5. Here, a = 1,a = a = a = a =
i1i2i3 1 2 3 111 233 232 223 323
a = a = 1,a = a = a = a = a = a = 1, and the other
332 322 3 145 154 451 415 541 514 2
elements of A are zero.
G
Definition 3.3. Let G = (V,E) be a hypergraph. The degree, d(v), of a
vertex v ∈ V is the number of edges consist of v.
Let G = (V,E) be a hypergraph, where V = {v ,v ,...,v } and E =
1 2 n
{e ,e ,...,e }. Then, the degree of a vertex v is given by
1 2 k i
n
d(v ) = a .
i ii2i3...im
i2,i3X,...,im=1
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Definition 3.4. A hypergraph is called k-regular if every vertex has the
same degree k.
Now, we discuss some spectral properties of A of a hypergraph G. Some
G
of these properties are very similar as in general graph (i.e. for a 2-uniform
hypergraph).
Theorem 3.1. Let µ be a H-eigenvalue of A . Then |µ| ≤ ∆, where ∆ is
G
the maximum degree of G.
Proof. Let G be a hypergraph of order m and dimension n. Let µ be a H-
eigenvalue of A = (a ) with an eigenvector x = (x ,x ,...,x ). Let
G i1i2...im 1 2 n
x = max{|x |,|x |,...,|x |}. Without loss of any generality we can assume
p 1 2 n
that x = 1. Now,
p
n
|µ| = |µxm 1| = a x x ...x
p − (cid:12) pi2i3...im i2 i3 im(cid:12)
(cid:12)(cid:12)i2,i3X,...,im=1 (cid:12)(cid:12)
(cid:12) (cid:12)
n
≤ |a ||x |m 1 = d(v ) ≤ ∆.
pi2i3...im p − p
i2,i3X,...,im=1
Thus, for a k-regular hypergraph the theorem (3.1) implies |µ| ≤ k.
Theorem 3.2. Let G = (V,E) be a k-regular hypergraph with n vertices.
Then, A = (a ) has a H-eigenvalue k.
G i1i2...im
Proof. Since, G is k-regular, then d(v ) = k for all v ∈ V, i ∈ {1,2,3,...,n}.
i i
Now, for a vector x = (1,1,1,...,1) ∈ Rn we have
n
A xm 1 = a = k.
G − ii2i3...im
i2,i3X,...im=1
Thus the proof.
Theorem 3.3. Let G = (V,E) be a k-regular hypergraph with n vertices.
Then, A = (a ) has a Z-eigenvalue k( 1 )m 2.
G i1i2...im √n −
Proof. Thevector x = ( 1 , 1 ,..., 1 ) ∈ Rn satisfies theZ-eigenvalueequa-
√n √n √n
tions for λ = k( 1 )m 2.
√n −
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Theorem 3.4. Let G be a hypergraph with n vertices andmaximum degree∆.
Let x = (x ,x ,...,x ) be a Z-eigenvector of A = (a ) corresponding
1 2 n G i1i2...im
to an eigenvalue µ. If x = max |x |,|x |,...,|x | , then |µ| ≤ ∆.
p 1 2 n xp
(cid:8) (cid:9)
Proof. The Z-eigenvalue equations of A for µ and x are Axm 1 = µx, and
G −
x2 = 1. Therefore, |x | ≤ 1, for all i = 1,2,3,...,n. Now,
i i
P
n
|µ||x | = a x x ...x ,
j (cid:12) ji2i3......im i2 i3 im(cid:12)
(cid:12)(cid:12)i2,i3X,...,im=1 (cid:12)(cid:12)
(cid:12) (cid:12)
which implies |µ||x | ≤ d(j) ≤ ∆,∀j = 1,2,3,...,n. Therefore, |µ| ≤ ∆.
j xp
Definition 3.5. A hypergraph H = (V ,E ) is said to be a spanning subhy-
1 1
pergraph of a hypergraph G = (V,E), if V = V and E ⊆ E.
1 1
′ ′
Theorem 3.5. Let G = (V,E) be hypergraph. Let H = (V ,E ) be a subhy-
pergraph of G, such that, m.c.e(G) = m.c.e(H) be even. Then, µ (H) ≤
max
µ (G), where µ is the highest Z-eigenvalue of the corresponding adja-
max max
cency hypermatrix.
′ ′
Proof. Let |V| = n, |V | = n (≤ n) and m.c.e(G) = m.c.e(H) = m. Now,
µ (H) = max xtA xm 1 (by using lemma (3.1) in [12])
max x m=1 H −
|| ||
′
n
= max aH x x ...x
||x||m=1(cid:18) i1i2...im i1 i2 im(cid:19)
i1,i2X,...,im=1
n
= max aH x x ...x , where aH = x = 0 when i > n′
||x||m=1(cid:18) i1i2...im i1 i2 im(cid:19) i1..im ir r
i1,i2X,...,im=1
n
≤ aG x x ...x
(cid:18) i1i2.....im i1 i2 im(cid:19)
i1,i2X,...,im=1
≤ µ (G),
max
since the each components of x are nonnegative (by Perron-Frobenious theo-
rem [3]) and the number of edges of G is greater than or equal to the number
edges of H. Hence the proof.
Definition 3.6. Let G = (V,E) be a hypergraph with V = {v ,v ,...,v },
1 2 n
E = {e ,e ,...,e }, and m.c.e(G) = m. Let x = (x ,x ,...,x ) be a vector
1 2 k 1 2 n
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in Rn and p ≥ s−1 be an integer. For an edge e = {v ,v ,...,v } and a
l1 l2 ls
vertex v , we define
li
xe/vli := x x ...x ,
p r1 r2 rp
X
where the sum is over r ,r ,...,r are chosen in all possible way from
1 2 p
{l ,l ,...,l }, such that, all l (j 6= i) occur at least once. Where as,
1 2 s j
xe := x x ...x ,
p r1 r2 rp
X
where the sum is over r ,r ,...,r are chosen in all possible way from
1 2 p
{l ,l ,...,l } with at least once for each element of the set.
1 2 s
The symmetric (adjacency) hypermatrix A of order m and dimension n
G
uniquely defines a homogeneous polynomial of degree m and in n variables
by
n
F (x) = a x x ...x .
G i1i2...im i1 i2 im
A
i1,i2X,,...im=1
We rewrite the above polynomial as:
F (x) = aGxe ,
G e m
A
X
e E
∈
where aG = s, α = m! , and s is the cardinality of
the edgeee . α Pk1,k2,...,ks≥1,Pki=m k1!k2!...ks!
Definition 3.7. Let G and H be two hypergraphs. The cartesian product,
G×H, of G and H is defined by the vertex set V(G×H) = V(G)×V(H)
and the edge set E(G×H) = {v}×e : v ∈ V(G),e ∈ E(H) e×{v} :
e ∈ E(G),v ∈ V(H) . (cid:8) (cid:9)S(cid:8)
(cid:9)
Definition3.8. LetGbeahypergraphwiththevertexsetV = {v ,v ,...,v }
1 2 n
and m.c.e(G) = m. For an edge e = {v ,v ,...,v } and integer r ≥ m the
l1 l2 ls
arrangement, (v v ...v ) (where p ,p ,...,p are chosen in all possible
p1 p2 pr 1 2 r
way from {l ,l ,...,l } with at least once for each element of the set) repre-
1 2 s
sents the edge e in order r.
Example 3.3. Let G = (V,E) whereV = {1,2,3,4,5}andE = {1,2,3},{2,3,5},{1,3,4,5} ,
then the arrangement(12233) represents the edge {1,2,3} in orde(cid:8)r 5. (12123) (cid:9)
is also a representation of the edge {1,2,3} in order five, whereas, (111123)
represents the edge {1,2,3} in 6 order.
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Let G = (V,E) be a hypergraph with m.c.e(G) = m and E = {e ∈ E :
i
v ∈ e}. Now, the H-eigenvalue equation for A becomes
i G
aGxe/vi = λx(m−1), for all i.
e m 1 i
X −
e Ei
∈
Theorem 3.6. Let G and H be two hypergraphs with m.c.e(G) = m.c.e(H).
If λ and µ are H-eigenvalue for G and H, respectively, then λ + µ is a
H-eigenvalue for G×H.
Proof. Let n and n be the number of vertices in G and H, respectively,
1 2
and m.c.e(G) = m.c.e(H) = m. Let (λ,u) and (µ,v) be H-eigenpairs of A
G
and A , respectively. Let w ∈ Cn1n2 be a vector with the entries indexed
H
by the pairs (a,b) ∈ [n ]×[n ], such that, w(a,b) = u(a)v(b). Now, we show
1 2
that (λ+µ,w) is an H-eigenpair of A .
G H
×
aGe×Hwme/(a1,b) = aGe×Hwm{a}×1e/(a,b) + aGe×Hwme×{1b}/(a,b)
X − X − X −
e∈E(a,b) {a}×e∈E(a,b) e×{b}∈E(a,b)
with e∈Eb with e∈Ea
= aG Hum 1(a)ve/b + aG Hue/a vm 1(b)
e× − m 1 e× m 1 −
X − X −
e∈Hb e∈Ga
= um 1(a) aHve/b +vm 1(b) aGue/a
− e m 1 − e m 1
X − X −
e∈Hb e∈Ea
= um 1(a)µvm 1(b)+vm 1(b)λum 1(a)
− − − −
= (λ+µ)wm 1(a,b).
−
Hence the proof2.
Lemma 3.1. Let A and B be two symmetric hypermatrix of order m and
dimension n, where m is even. Then λ (A + B) ≤ λ (A) + λ (B),
max max max
where λ (A) denotes the largest Z-eigenvalue of A.
max
Proof.
λ (A+B) = max xt(A+B)xm 1 (by using lemma (3.1) in [12])
max x m=1 −
|| ||
≤ max xtAxm 1 +max xtBxm 1
x m=1 − x m=1 −
|| || || ||
= λ (A)+λ (B).
max max
2For the similar proof on uniform hypergraph see [7].
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Let G = (V,E) be a hypergraph with the vertex set V = {v ,v ,...,v }
1 2 n
and m = max{|e | : e ∈ E}. We partition the edge set E as, E = E ∪E ∪
i i 1 2
···∪E , where E contains all the edges of the cardinality i and construct
m i
a hypergraph G = (V,E ), for a nonempty E .
i i i
Definition 3.9. Define the adjacency hypermatrix of G in m (> i) -order
i
by a n dimensional m order hypermatrix
Am = (am) , 1 ≤ p ,p ,...,p ≤ n,
Gi Gi p1p2...pm 1 2 m
(cid:0) (cid:1)
such that, for any e = {v ,v ,...,v } ∈ E ,
l1 l2 li i
i m!
(am) = , where α =
Gi p1p2...pm α k !k !...k !
X 1 2 i
k1,k2,...,ki 1,Pkj=m
≥
and p ,p ,...,p are chosen in all possible way from {l ,l ,...,l } with at
1 2 m 1 2 i
least once for each element of the set. The other positions of Am are zero.
Gi
Thus, we can represent a hypergraph G, with m.c.e(G) = s, in higher
order m > s by the hypermatrix Am. Clearly, all the eigenvalue equations
G
show that the eigenvalues of Am1 and Am2 are not equal for m 6= m .
G G 1 2
Theorem 3.7. Let G = (V,E) be a hypergraph and m.c.e(G) = m be
even. Then λ (A ) ≤ m λ (Am), where λ (A) is the largest Z-
max G i=1 max Gi max
eigenvalue of A. P
Proof. Since A = m Am, the proof follows from the lemma (3.1).
G i=1 Gi
P
Moreover, thetheorem(3.7)impliesλ (A ) ≤ m n λ (Am),where
max G i=1 i max i
ni is the number of edges of cardinality i and Ami isPthe adjacency hyperma-
trix in m-order of a hypergraph contains a single edge of cardinality i.
3.2 Laplacian hypermatrix and eigenvalues
Definition 3.10. Let G = (V,E) be a (general) hypergraph without any
isolated vertex where V = {v ,v ,...,v } and E = {e ,e ,...,e }. Let
1 2 n 1 2 k
m.c.e(G) = m. DefinetheLaplacianhypermatrix, L , ofG = (V,E)asL =
G G
D −A = (l ) where 1 ≤ i ,i ,...,i ≤ n, where D = (d ) is
G G i1i2...im 1 2 m G i1i2...im
the m order n dimensional diagonal hypermatrix with d = d(v ) and
ii...i i
others are zero. The signless laplacian of G is defined as L = D +A .
G G G
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