Spectra of General Hypergraphs: By using tensor › ~ncap2016 › sites › default › ... · broadly used in graphs and adapted to the hypergraphs. This is a compilations of several

Tensors related to a HypergraphEigenvalues of a Tensor

Tensors for Non-uniform Hypergraphs and Their Eigenvalues

Spectra of General Hypergraphs: By using tensor

Anirban Banerjee

Network Theory: Conceptual Advances and Practical ApplicationsIMSc, Chennai

26 April 2016

A. Banerjee Hypergraph spectra



Graph

Graph G = (V ,E )

V ≡ Vertex set

E ≡ Edge set ⊆ V × V

1

2

4

3

Graph G = (V ,E )

V = {1, 2, 3, 4}E = {12, 13, 23, 34}




Networks Represented by GraphsExamples

Component of a system ⇒ Vertex (Node)Interaction or relation between two components ⇒ Edge

1 Film actor network

An actor / actress ⇒ VertexIf two of them acted in a movie ⇒ Edge

2 Co-authorship network

An author ⇒ VertexIf two authors have written an article jointly ⇒ Edge

3 Protein-protein interaction network

Protein ⇒ VertexDirect physical interactions (binding) ⇒ Edge

4 Metabolic network

Metabolites ⇒ VertexIf two metabolites take part in the same reaction ⇒ Edge




Hypergraph

Hypergraph G = (V ,E )

V ≡ Vertex set

E ≡ Edge set ⊆ power setof V

Hypergraphs: a bioinformatics reviewPablo Vicente-Munuera

Bioinformatics MScEmail: [email protected]

Abstract—The aim of this work is to give an overview of thestate of the art techniques involving hypergraphs at the field ofbioinformatics and computational biology. Also, we review severalinteresting works about hypergraphs and show the advantagesand disadvantages of using them, instead of graphs.

I. INTRODUCTION

New perspectives, ’new’ solutions to antique problems anda whole new perspective, which has his own problems. That’swhat we’re getting when we deal with hypergraphs. In almostevery recent paper, we see, in the first few pages, the mathsformulation of the hypergraph. We are, right now, at the firststage of this technology, so that’s the usual way to presenta new whole framework. Promising framework, indeed. But,when we talk about ’new’, it’s not the idea itself, we talk aboutapplications, and model old problems into this solutions, nevermade before. About the idea of hypergraph, it has a ’few’years, like all the maths we use and revolutionized the world.The zenith came in the mid 80’s with the two editions of thebasics of hypergraphs with Bollobas et al and Berge et al in[2] and [3], respectively. The two of them are essentials inthese days. Besides history, what’s a hypergraph? And whyis a ’revolution’? Let’s answer these questions in the nextparagraphs.

A. What’s a hypergraph?

A hypergraph is a generalization of a graph. Or, said inother words, a graph is a particular case of the hypergraph.A specific case, in which all the hyperedges are incident withtwo vertices (whichever vertices), so, we’ll have the so callededges (or, in the directed case, an arc).

Therefore, and formally, a hypergraph H is a pair H =(V, E) where V are a set of finites vertices and E is a non-empty set of hyperedges. The definition of a hyperedge ispretty clear here: it’s a generalized edge that could connectsince one vertex to |V |, or the existing vertices. As we cansee in the fig. 1, the hyperedge e4 is only connected to onevertex (v4), and, in the other hand, we have e1 a hyperedgeinvolving the vertices v1, v2 and v3. These are examples ofhyperedges and some hypergraph.

B. Is it really a revolution?

Clear the definition of a hypergraph, we have to tackleother questions. Is this revolution? If we talk mathematically,absolutely not. As said in the introduction, the term was mintedaround 50’s, and it has been used since then as a part ofcombinatorics.

Fig. 1. An example of hypergraph. Thanks to [4]

However, since the biology explode in a synthesis of awhole new world like bioinformatics, there’s an existing needof model new data and try to see different things. This lastconcept of ”see different things” is key of why hypergraphsare, indeed, a revolution. In order to explain this, I’m going totalk about graphs. When you represent something in a graphyou are representing relationships between one object withother. And, when you analyze the whole graph with somemeasure like betweenness or random walks, you are, actually,measuring this 1-on-1 relations and seen this as a whole.Therefore, this model of graphs gives you simpler interpreta-tions, and you only could simulate that kind of relationships,like peer to peer or a social network involving one personconnecting other one. But, what if you have a system that can’tbe modelled by 1-on-1 situations? Like chemical reactions (fig.2), or miRNA-mRNA networks like in [6], and many othersshown in [11]. If you chose graph as your solution, you’dprobably lose information. And, losing information, valuableinformation, doesn’t have to an be option1.

The revolution become when you realize a hypergraphcapture multiple relationships among features. Thus, someauthors like Hancock et al in [7] calls this process, high orderlearning due to what you could appreciate at the graph, arehigher order relationships.

II. METHODS

This section would provide you some classic measuresbroadly used in graphs and adapted to the hypergraphs. Thisis a compilations of several papers like [18], [10] and [12].

A. Sub-hypergraph centrality

Since ’whole-graph’ centrality would have too much com-plexity, you could use sub-hypergraph centrality. So we definethis centrality of a vertex v, as the summation of the closed

1Probably in other cases like when you do a PCA, you’re losing informa-tion, but you’re reducing the noise, so this case is different.

Hyperraph G = (V ,E )

V = {v1, v2, v3, v4, v5, v6, v7}E ={{v1, v2, v3}, {v2, v3}, {v3, v5, v6}, {v4}}

”Hypergraph-wikipedia” by Hypergraph.svg: Kilom691derivative work: Pgdx (talk) - Hypergraph.svg. Licensed under CC BY-SA 3.0 via

Wiki-media Commons - https://commons.wikimedia.org/wiki/File:Hypergraph- wikipedia.svg/media/File:Hypergraph-wikipedia.svg




Networks Represented by HypergraphsExamples

Component of a system ⇒ Vertex (Node)

Interaction or relation between two× components ⇒ Edge.

1 Film actor network

Vertex ⇒ An actor / actressEdge ⇒ Cast of a film

2 Co-authorship network

Vertex ⇒ An authorEdge ⇒ All the authors of an article

3 Protein-protein interaction network

Vertex ⇒ ProteinEdge ⇒ All the proteins in a protein complex

4 Metabolic network

Vertex ⇒ MetabolitesEdge ⇒ All the metabolites take part in a reaction




Networks Represented by HypergraphsExamples

node participates in) equal to or largerthan k. The maximum core of a graphcorresponds to the highest k where thegraph has a non-empty k-core. Themaximum k-core of the graph inFigure 1A is a 3-core consisting of thenodes {A,B,C,D}. A similar definition of ak-core can be defined for (Sperner) hyper-graphs, where k corresponds to thenumber of hyperedges each node partic-

ipates in [5]. The maximum k-core of thehypergraph in Figure 1A is a 2-coreconsisting of {A,C,E}. Thus, as one wouldintuitively expect, the maximum k-core ofthe hypergraph ranks A, C, and E as mostimportant—in contrast to the graph mod-el, whose maximum k-core would weight Band D stronger than E.Another application of undirected hy-

pergraphs is minimal hitting sets (MHSs), also

known as generalized vertex covers orhypergraph transversals [6,7]. For exam-ple, in a given hypergraph model of a PPInetwork, an interesting problem related toexperimental design [5] is to determineminimal (irreducible) subsets of bait pro-teins that would cover or ‘‘hit’’ allcomplexes in a minimal way; i.e., noproper subset of an MHS would hit allcomplexes. In Figure 1A, the correspond-

Figure 1. Examples of undirected (A,B) and directed (C,D) hypergraphs arising in the context of biological networks analysis.Detailed explanations are given in the text.doi:10.1371/journal.pcbi.1000385.g001

PLoS Computational Biology | www.ploscompbiol.org 2 May 2009 | Volume 5 | Issue 5 | e1000385

node participates in) equal to or largerthan k. The maximum core of a graphcorresponds to the highest k where thegraph has a non-empty k-core. Themaximum k-core of the graph inFigure 1A is a 3-core consisting of thenodes {A,B,C,D}. A similar definition of ak-core can be defined for (Sperner) hyper-graphs, where k corresponds to thenumber of hyperedges each node partic-

ipates in [5]. The maximum k-core of thehypergraph in Figure 1A is a 2-coreconsisting of {A,C,E}. Thus, as one wouldintuitively expect, the maximum k-core ofthe hypergraph ranks A, C, and E as mostimportant—in contrast to the graph mod-el, whose maximum k-core would weight Band D stronger than E.Another application of undirected hy-

pergraphs is minimal hitting sets (MHSs), also

known as generalized vertex covers orhypergraph transversals [6,7]. For exam-ple, in a given hypergraph model of a PPInetwork, an interesting problem related toexperimental design [5] is to determineminimal (irreducible) subsets of bait pro-teins that would cover or ‘‘hit’’ allcomplexes in a minimal way; i.e., noproper subset of an MHS would hit allcomplexes. In Figure 1A, the correspond-

Figure 1. Examples of undirected (A,B) and directed (C,D) hypergraphs arising in the context of biological networks analysis.Detailed explanations are given in the text.doi:10.1371/journal.pcbi.1000385.g001

PLoS Computational Biology | www.ploscompbiol.org 2 May 2009 | Volume 5 | Issue 5 | e1000385

=⇒ Protein-Protein InteractionNetwork

=⇒ Metabolic ReactionsNetwork

Hypergraphs and Cellular Networks, PLoS Comput Biol, 5(5), 2009




Graph and Connectivity MatrixAdjacency matrix

1

2

4

3

Graph G = (V ,E )

V = {1, 2, 3, 4}E = {12, 13, 23, 34}

A =

0 1 1 01 0 1 01 1 0 10 0 1 0

4×4

12 ∈ E ⇒A12 = 1A21 = 1

Degree of vertex 1 =∑

j A1j




Hypergraph and Connectivity MatrixAdjacency matrix

Hypergraphs: a bioinformatics reviewPablo Vicente-Munuera

Bioinformatics MScEmail: [email protected]

Abstract—The aim of this work is to give an overview of thestate of the art techniques involving hypergraphs at the field ofbioinformatics and computational biology. Also, we review severalinteresting works about hypergraphs and show the advantagesand disadvantages of using them, instead of graphs.

I. INTRODUCTION

New perspectives, ’new’ solutions to antique problems anda whole new perspective, which has his own problems. That’swhat we’re getting when we deal with hypergraphs. In almostevery recent paper, we see, in the first few pages, the mathsformulation of the hypergraph. We are, right now, at the firststage of this technology, so that’s the usual way to presenta new whole framework. Promising framework, indeed. But,when we talk about ’new’, it’s not the idea itself, we talk aboutapplications, and model old problems into this solutions, nevermade before. About the idea of hypergraph, it has a ’few’years, like all the maths we use and revolutionized the world.The zenith came in the mid 80’s with the two editions of thebasics of hypergraphs with Bollobas et al and Berge et al in[2] and [3], respectively. The two of them are essentials inthese days. Besides history, what’s a hypergraph? And whyis a ’revolution’? Let’s answer these questions in the nextparagraphs.

A. What’s a hypergraph?

A hypergraph is a generalization of a graph. Or, said inother words, a graph is a particular case of the hypergraph.A specific case, in which all the hyperedges are incident withtwo vertices (whichever vertices), so, we’ll have the so callededges (or, in the directed case, an arc).

Therefore, and formally, a hypergraph H is a pair H =(V, E) where V are a set of finites vertices and E is a non-empty set of hyperedges. The definition of a hyperedge ispretty clear here: it’s a generalized edge that could connectsince one vertex to |V |, or the existing vertices. As we cansee in the fig. 1, the hyperedge e4 is only connected to onevertex (v4), and, in the other hand, we have e1 a hyperedgeinvolving the vertices v1, v2 and v3. These are examples ofhyperedges and some hypergraph.

B. Is it really a revolution?

Clear the definition of a hypergraph, we have to tackleother questions. Is this revolution? If we talk mathematically,absolutely not. As said in the introduction, the term was mintedaround 50’s, and it has been used since then as a part ofcombinatorics.

Fig. 1. An example of hypergraph. Thanks to [4]

However, since the biology explode in a synthesis of awhole new world like bioinformatics, there’s an existing needof model new data and try to see different things. This lastconcept of ”see different things” is key of why hypergraphsare, indeed, a revolution. In order to explain this, I’m going totalk about graphs. When you represent something in a graphyou are representing relationships between one object withother. And, when you analyze the whole graph with somemeasure like betweenness or random walks, you are, actually,measuring this 1-on-1 relations and seen this as a whole.Therefore, this model of graphs gives you simpler interpreta-tions, and you only could simulate that kind of relationships,like peer to peer or a social network involving one personconnecting other one. But, what if you have a system that can’tbe modelled by 1-on-1 situations? Like chemical reactions (fig.2), or miRNA-mRNA networks like in [6], and many othersshown in [11]. If you chose graph as your solution, you’dprobably lose information. And, losing information, valuableinformation, doesn’t have to an be option1.

The revolution become when you realize a hypergraphcapture multiple relationships among features. Thus, someauthors like Hancock et al in [7] calls this process, high orderlearning due to what you could appreciate at the graph, arehigher order relationships.

II. METHODS

This section would provide you some classic measuresbroadly used in graphs and adapted to the hypergraphs. Thisis a compilations of several papers like [18], [10] and [12].

A. Sub-hypergraph centrality

Since ’whole-graph’ centrality would have too much com-plexity, you could use sub-hypergraph centrality. So we definethis centrality of a vertex v, as the summation of the closed

1Probably in other cases like when you do a PCA, you’re losing informa-tion, but you’re reducing the noise, so this case is different.


V = {v1, v2, v3, v4, v5, v6, v7}E = {{v1, v2, v3}, {v2, v3},{v3, v5, v6}, {v4}}

A =?




Uniform Hypergraph and Connectivity TensorAdjacency tensor

1

2

3

4

5


V = {1, 2, 3, 4, 5}E ={{1, 2, 3}, {2, 3, 4}, {3, 4, 5}}

General graph ≡ 2-uniform hypergraph

A5×5×5 ⇒ Adjacency tensor oforder 3

{1, 2, 3} ∈ E ⇒A123 = 1/k,A132 = 1/k,A213 = 1/k,A231 = 1/k,A312 = 1/k,A321 = 1/k,

Degree of vertex 1 =∑

j1j2A1j1j2

k = 1/(3− 1)!




Adjacency Matrix and Adjacency Tensor of m-UniformHypergraph

Let G = (V ,E ) be a graph / hypergraph where V = {v1, v2, ..., vn} andE = {e1, e2, ..., ek}

GraphAi1i2 ={

1, if (i1, i2) ∈ E

0, otherwise.

m-uniform HypergraphAi1i2...im ={

1/(m − 1)!, if {vi1 , . . . , vim} ∈ E

0, otherwise.

Degree of vertex vi is d(vi ) =∑n

i2,...,ir=1 Aii2...im




Laplacian Matrix and Laplacian Tensor of m-UniformHypergraph

GraphL = D − A

Dii := d(vi ) is a diagonal matrix

m-uniform HypergraphL = D −A

Dii...i := d(vi ) is a diagonal tensorof order m.




Signless Laplacian Matrix and Signless Laplacian Tensor ofm-Uniform Hypergraph

GraphL = D + A

Dii := d(vi ) is a diagonal matrix

m-uniform HypergraphL = D + A

Dii...i := d(vi ) is a diagonal tensorof order m .




Normalized Laplacian Matrix and Normalized LaplacianTensor of m-Uniform Hypergraph

Graph

Li1i2 =

−1√

d(vi1 )d(vi2 ), if (vi1 , vi2) ∈ E

1, if i1 = i2, d(vi1) 6= 0,

0, otherwise.

m-uniform Hypergraph

Li1..im =

−1

(m−1)!∏m

j=11

m√

d(vij ), if {vi1 , .., vim} ∈ E

1, if i1 = ... = im, d(vi1) 6= 0,

0, otherwise.




Normalized Laplacian Matrix and Normalized LaplacianTensor of m-Uniform Hypergraph

Graph

Li1i2 =

−1

d(vi1 ), if (vi1 , vi2) ∈ E

1, if i1 = i2, d(vi1) 6= 0,

0, otherwise.

m-uniform Hypergraph

Li1..im =

−1

(m−1)!1

d(vi1 ), if {vi1 , .., vim} ∈ E

1, if i1 = ... = im, d(vi1) 6= 0,

0, otherwise.




Tensor Product

C order (m − 1)(k − 1) + 1 = A order (m ≥ 2) · B order (k ≥ 1)

Ciα1..αm−1 :=n∑

i2..im=1

Aii2..imBi2α1 ..Bi2αm−1 (i ∈ [n], α1, .., αm−1 ∈ [n−1]k−1)

All of them have the same dimension n.

If A (or B) is a matrix, C has the same order as B (or A).

If B = x is a vector of dimension n,

Ci = (A · B)i = (A · x)i =n∑

i2..im=1

Aii2..imxi2 ..xim .




Tensor Operators (Order m)

(A · x)i1 =1

(m − 1)!

n∑i2..im=1,(i1i2..im)∈E

xi2 ..xim .

(L · x)i1 =1

(m − 1)!

n∑i2..im=1,(i1i2..im)∈E

(xm−1i1− xi2 ..xim)

= d(vi1)xm−1i1− 1

(m − 1)!

n∑i2..im=1,(i1i2..im)∈E

xi2 ..xim .

(L · x)i1 = xm−1i1− 1

(m − 1)!

1

d(vi1)

n∑i2..im=1,(i1i2..im)∈E

xi2 ..xim .

(L · x)i1 = xm−1i1− 1

(m − 1)!

n∑i2..im=1,(i1i2..im)∈E

m∏j=1

1m√d(vij )

xi2 ..xim .




H-eigenpair

(λ, x) ∈ C× (Cn \ {0}) is called eigenvalue and eigenvector (or simply aneigenpair) if

A · x = λx [m−1], where x [m] = (xm1 , ..., xmn )

(λ, x) ∈ R× (Rn \ {0})⇒ H-eigenpair.

(λ, x) ∈ R× (Rn+ \ {0})⇒ H+-eigenpair.

(λ, x) ∈ R× (Rn++ \ {0})⇒ H++-eigenpair.




E -eigenpair and Z -eigenpair

(λ, x) ∈ C× (Cn \ {0})⇒ E -eigenpair if

A · x = λx ,

n∑i=1

x2i = 1.

if (λ, x) ∈ R× (Rn \ {0})⇒ Z -eigenpair.




Matrix-Tensor-Matrix Product

A is a tensor of order m and dimension n. P,Q are matrices.

(P · A · Q)i1...im =n∑

j1..jm=1

Aj1..jmPi1j1Qj2i2 ..Qjm im




Similar Tensor

P,Q are matrices. A,B are tensors of order m and dimension n. A,B aresimilar tensors, i.e. have the same eigenvalues if

B = P · A · Q,

P · I · Q = I where I is unit tensor.

A and B are diagonal similar if B = D−(m−1) · A · D,, for anyinvertible diagonal matrix D.

A and B are permutational similar if for a permutation matrix P,B = P · A · PT

⇒ Isomorphic hypergraphs are isospectral.




Tensors for Non-uniform Hypergraphs

How to represent a non-uniform hypergraphby a tensor?




Adjacency Tensors for Non-uniform Hypergraphs

Hypergraph G = (V ,E ), V = {v1, v2, . . . , vn}, E = {e1, e2, . . . , ek},m.c .e(G ) = m = max{|ei | : ei ∈ E}.For all edges e = {vl1 , vl2 , . . . , vls} ∈ E of cardinality s ≤ m,

Ap1p2...pm =1

α/s, where α =

∑k1,k2,...,ks≥1,

∑ki=m

m!

k1!k2! . . . ks !,

p1, p2, . . . , pm are chosen in all possible way from {l1, l2, . . . , ls} with atleast once for each element of the set. The other positions of A are zero.

d(vi ) =n∑

i2,i3,...,im=1

Aii2i3...im .




Adjacency Tensors for Non-uniform HypergraphsAn example

G = (V ,E ), V = {1, 2, 3, 4, 5}, E ={{1}, {2, 3}, {1, 4, 5}

}.

Here, m.c .e(G ) = m = 3.

A111 = 1,

A233 = A232 = A223 = A323 = A332 = A322 = 13 ,

A145 = A154 = A451 = A415 = A541 = A514 = 12 ,

the other elements of A are zero.

d(v1) = 2, d(v2) = 1.




Adjacency Tensors for Non-uniform HypergraphsEigenvalue properties

µ ≡ H-eigenvalue of AG ⇒

|µ| ≤ max. deg(G ).

(µ, x) ≡ Z -eigenpair of AG , x = (x1, x2, . . . , xn)⇒

|µ| ≤ max. deg(G )

max{|x1|, |x2|, . . . , |xn|

} .A k-regular hypergraph with n vertices

- has a H-eigenvalue k,

- has a Z -eigenvalue k( 1√n

)m−2.





λ ≡ H-eigenvalue (G ), µ ≡ H-eigenvalue (H),m.c .e(G ) = m.c .e(H)⇒

λ+ µ ≡ H-eigenvalue (G × H).

H ⊆ G , m.c .e(G ) = m.c .e(H) ≡ even ⇒

max. Z -eigenvalue (H) ≤ max. Z -eigenvalue (G ).





G = (V ,E ), m.c .e(G ) = m ≡ even.Gi = (V ,Ei ), Ei = {e ∈ E : |e| = i} 6= Φ.

Adjacency tensor of Gi in order m ≥ i ≡ AmGi

For any e = {vl1 , . . . , vli} ∈ Ei ,

(AmGi

)p1p2...pm = 1α/i , where α =

∑k1,k2,...,ki≥1,

∑kj=m

m!k1!k2!...ki !

p1, p2, . . . , pm are chosen in all possible way from {l1, l2, . . . , li} withat least once for each element of the set. The other positions of Am

Gi

are zero. =⇒

max. Z -eigenvalue (AG ) ≤m∑i=1

max. Z -eigenvalue (AmGi

).




Laplacian Tensors for Non-uniform Hypergraphs

G = (V ,E ),

L = D −A,

Dii...i := d(vi ) is a diagonal tensor of order m = m.c .e(G ).

L is not positive semidefinite, but, copositive tensor,i.e. Lxm ≥ 0 ∀x ∈ Rn

+.




Laplacian Tensors for Non-uniform HypergraphsEigenvalue properties

G = (V ,E ),m.c .e(G ) ≥ 3. Then

L has an H-eigenvalue 0 with eigenvector (1, 1, . . . , 1) ∈ Rn.

0 is the unique H++ eigenvalue of L.

L has an Z -eigenvalue 0 with the same eigenvector.





G = (V ,E ),m.c .e(G ) ≥ 3. Then

(d(i), e(j)) is an H-eigenpair, where e(j) ∈ Rn and e(j)i = 1 if i = j ,

otherwise 0.

max. H+-eigenvalue (LG ) = max. Degree (G ).





G = (V ,E ),

0 ≤ H-eigenvalue(LG ) ≤ 2.max. Degree (G ).

G is connected iffmin

j=1,...,nmin{Lxm|x ∈ Rn

+,∑n

i=1 xmi = 1, xj = 0} > 0.




Normalized Laplacian Tensors for Non-uniformHypergraphs

Hypergraph G = (V ,E ), V = {v1, v2, . . . , vn}, E = {e1, e2, . . . , ek},m.c .e(G ) = m = max{|ei | : ei ∈ E}.For all edges e = {vl1 , vl2 , . . . , vls} ∈ E ,

Lp1p2...pm = − 1

α/s

m∏j=1

1m√

d(vpj ), where α =

∑k1,k2,...,ks≥1,

∑ki=m

m!

k1!k2! . . . ks !,

p1, p2, . . . , pm are chosen in all possible way from {l1, l2, . . . , ls} with atleast once for each element of the set. The diagonal entries of L are 1and the rest are zero.




Normalized Laplacian Tensors for Non-uniformHypergraphs

For all edges e = {vl1 , vl2 , . . . , vls} ∈ E of cardinality s ≤ m,

Lp1p2...pm = − 1

d(vp1)

1

α/s, where α =

∑k1,k2,...,ks≥1,

∑ki=m

m!

k1!k2! . . . ks !,

p1, p2, . . . , pm are chosen in all possible way from {l1, l2, . . . , ls} suchthat, all lj occur at least once. All the diagonal entries are 1 and the restare zero.

Normalized adjacency tensor ≡ A = I− L⇒ Stochastic tensor.




Normalized Laplacian Tensors for Non-uniformHypergraphsEigenvalue properties

L and L are diagonal similar ⇒ co-spectral.

G = (V ,E ),E 6= Φ.λ ∈ σ(A)⇔ (1− λ) ∈ σ(L), otherwise, σ(A) = σ(L) = 0, σ(L) ≡spectrum of L.

G = (V ,E ) has r ≥ 1 connected components, G1,G2, . . .Gr ,|V (Gi )| > 1,m.c .e(Gi ) = m.c .e(G )∀i . Then, as sets,σ(A) = σ(A1) ∪ σ(A2) ∪ · · · ∪ σ(Ar ).




Normalized Laplacian Tensors for Non-uniformHypergraphsEigenvalue properties

ρ(A) = 1,

0 ≤ λ(L) ≤ 2,

0 is an eigenvalue of L with the eigenvector {1, 1, . . . 1}

0 is the unique H++ eigenvalue of L .




THANK YOU!

Ref: A. Banerjee, A. Char, B. Mondal. Spectra of general hypergraphs. Preprint.

E-print: arXiv:1601.02136.


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Spectra of General Hypergraphs: By using tensor › ~ncap2016 › sites › default › ... · broadly used in graphs and adapted to the hypergraphs. This is a compilations of several