Intersection Graphs of Oriented Hypergraphs and Their Matrices
IINTERSECTION GRAPHS OF ORIENTED HYPERGRAPHS ANDTHEIR MATRICES
NATHAN REFF ∗ Abstract.
For a given hypergraph, an orientation can be assigned to the vertex-edge incidences.This orientation is used to define the adjacency and Laplacian matrices. In addition to studying thesematrices, several related structures are investigated including the incidence dual, the intersectiongraph (line graph), and the 2-section. A connection is then made between oriented hypergraphs andbalanced incomplete block designs.
Key words.
Oriented hypergraph, intersection graph, line graph, hypergraph adjacency matrix,hypergraph Laplacian, signed graph, signed hypergraph, balanced incomplete block designs.
AMS subject classifications.
1. Introduction. An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1 or −
1. Shi called this type of hypergraph a signed hypergraph and used it to model the constrained via minimization (CVM)problem or two-layer routings [25, 26]. Oriented hypergraphs were independentlydeveloped to generalize oriented signed graphs [27] and related matroid properties [22,24]. A generalization of directed graphs, known as directed hypergraphs , also have thistype of vertex-edge labeling (see for example [14], and the references therein). Whatdistinguishes oriented hypergraphs from these other related incidence structures is thenotion of an adjacency signature that naturally allows the adjacency and Laplacianmatrices to be defined and studied [22, 21, 5]. This is an alternative approach tostudying matrices and hypermatrices associated to hypergraphs [6, 9, 13, 18, 8, 17, 19],that does not require a uniformity condition on the edge sizes and allows reasonablyquick spectral calculations. Rodr´ıguez also developed a version of the adjacency andLaplacian matrices for hypergraphs without a uniformity requirement on edge sizes[23]. The definition of adjacency signature and the derived matrices could be appliedto directed hypergraphs and their many applications.This paper is a continuation of the investigation of matrices and eigenvaluesassociated to oriented hypergraphs and their related structures. Specifically, a varietyof intersection graphs of oriented hypergraphs are defined and algebraic relationshipsare found. ∗ Department of Mathematics, The College at Brockport: State University of New York, Brock-port, NY 14420, USA (nreff@brockport.edu). 1 a r X i v : . [ m a t h . C O ] S e p n Section 2 relevant background is provided. In Section 3 new oriented hy-pergraphs are defined, including the intersection graph (or line graph) of an orientedhypergraph. Some results on oriented hypergraphs that have particular signed graphsas their intersection graphs are shown. Sections 4 and 5 develop matrix and otheralgebraic relationships between an oriented hypergraph and its dual and intersectiongraphs. These matrix identities are then used to study the eigenvalues associated tothe adjacency and Laplacian matrices of the same incidence structures. In Section 6 aconnection between oriented hypergraphs, balanced incomplete block designs (BIBD)and their incidence matrices is given.In [3] the line graph of a directed hypergraph (called a line dihypergraph) is stud-ied. This construction extends the notion of the line digraph [16] to the hypergraphicsetting in order to study connectivity problems with applications to bus networks.The definition of the line graph of an oriented hypergraph that we will introduce inthis paper is quite different than the line dihypergraph. However, it would be aninteresting future project to see connections between these constructions.The reader may be interested in two other related investigations. A generalizationof the line digraph, called the partial line digraph, is defined in [12]. This too was gen-eralized to directed hypergraphs [11, 10] with connectivity and expandability results.Acharya studied signed intersection graphs [1], where an alternative hypergraphicversion of signed graphs is introduced.
2. Background.2.1. Oriented Hypergraphs.
Let Ω be a finite set of indexes. A hypergraph isa triple H = ( V, E, I ), where V is a set, E = ( e ι ) ι ∈ Ω is a family of subsets of V , and I is a multisubset of V × E such that if ( v, e ι ) ∈ I , then v ∈ e ι . Note that an edge maybe empty. The set V is called the set of vertices , and E is called the family of edges .We may also write V ( H ), E ( H ) and I ( H ) if necessary. Let n := | V | and m := | E | . If( v, e ) ∈ I , then v and e are incident . An incidence is a pair ( v, e ), where v and e areincident. If ( v i , e ) and ( v j , e ) both belong to I , then v i and v j are adjacent verticesvia the edge e .A hypergraph is simple if for every edge e , and for every vertex v ∈ e , v and e are incident exactly once. Unless otherwise stated, all hypergraphs in this paper areassumed to be simple. A hypergraph is linear if for every pair e, f ∈ E , | e ∩ f | ≤ degree of a vertex v i , denoted by d i = deg( v i ), is equal to the number ofincidences containing v i . The maximum degree is ∆( H ) = ∆ := max i d i . The size ofan edge e is the number of incidences containing e . A k -edge is an edge of size k . A k -uniform hypergraph is a hypergraph such that all of its edges have size k . The rank of H , denoted by r ( H ), is the maximum edge size in H . he incidence dual (or dual ) of a hypergraph H = ( V, E, I ), denoted by H ∗ , isthe hypergraph ( E, V, I ∗ ), where I ∗ := { ( e, v ) : ( v, e ) ∈ I} . Thus, the incidence dualreverses the roles of the vertices and edges in a hypergraph.The set of size 2 subsets of a set S is denoted by (cid:0) S (cid:1) . The set of adjacencies A of H is defined as A := { ( e, { v i , v j } ) ∈ E × (cid:0) V (cid:1) : ( v i , e ) ∈ I and ( v j , e ) ∈ I} .Observe that if { v i , v j } ∈ (cid:0) V (cid:1) , then the vertices v i and v j must be distinct. The setof coadjacencies A ∗ of H is defined as A ∗ := A ( H ∗ ).An oriented hypergraph is a pair G = ( H, σ ) consisting of an underlying hyper-graph H = ( V, E, I ), and an incidence orientation σ : I → { +1 , − } . Every orientedhypergraph has an associated adjacency signature sgn : A → { +1 , − } defined by(2.1) sgn( e, { v i , v j } ) = − σ ( v i , e ) σ ( v j , e ) . Thus, sgn( e, { v i , v j } ) is called the sign of the adjacency ( e, { v i , v j } ). Instead of writingsgn( e, { v i , v j } ), the alternative notation sgn e ( v i , v j ) will be used. The notation σ G may also be used for the incidence orientation when necessary. See Figure 2.1 for anexample of an oriented hypergraph. G G +1 +1 +1+1 +1 +1+1 − − Fig. 2.1 . A simple oriented hypergraph G drawn in two ways. On the left, the incidences arelabeled with σ values. On the right, the σ values assigned to the incidences are drawn using thearrow convention of +1 as an arrow going into a vertex and − as an arrow departing a vertex. As with hypergraphs, an oriented hypergraph has an incidence dual. The inci-dence dual of an oriented hypergraph G = ( H, σ ) is the oriented hypergraph G ∗ =( H ∗ , σ ∗ ), where the coincidence orientation σ ∗ : I ∗ → { +1 , − } is defined by σ ∗ ( e, v )= σ ( v, e ), and the coadjacency signature sgn ∗ : A ∗ → { +1 , − } is defined bysgn ∗ ( v, { e i , e j } ) = − σ ∗ ( e i , v ) σ ∗ ( e j , v ) = − σ ( v, e i ) σ ( v, e j ) . The notation σ G ∗ may also be used for the coincidence orientation. See Figure 2.2for an example of the dual. A signed graph is a graph where edges are givenlabels of either +1 or −
1. Formally, a signed graph Σ = (Γ , sgn) is a graph Γ togetherwith a sign function (or signature ) sgn : E (Γ) → { +1 , − } . An oriented signed graph ∗ Fig. 2.2 . The incidence dual G ∗ of G from Figure 2.1. (Σ , β ) is a signed graph together with an orientation β : I (Γ) → { +1 , − } that isconsistent with the sign function via the relation(2.2) sgn( e ij ) = − β ( v i , e ij ) β ( v j , e ij ) . Oriented signed graphs were developed [27] to generalize Greene’s bijection betweenacyclic orientations and regions of an associated hyperplane arrangement [15]. Thisgeneralization is intimately connected with the theory of oriented matroids [4].Notice that Equation 2.2 is the same formula used to calculate the adjacencysigns in an oriented hypergraph. This is merely because oriented signed graphs arethe model for the generalization of oriented hypergraphs. It should therefore be nosurprise that a 2-uniform oriented hypergraph is an oriented signed graph. If we saythat G is a simple oriented signed graph, then G has no loops or multiple edges. Putanother way, G is a 2-uniform linear oriented hypergraph.Another minor technical note: in order to create an oriented signed graph (Σ , β )for a given signed graph Σ, many different orientations β can be chosen so that Equa-tion 2.2 is satisfied . This freedom of choice does not usually cause any issues in thecalculations. There may, however, be specific situations when particular orientationsare more favorable and this will be noted. Let G = ( H, σ ) be an orientedhypergraph. The adjacency matrix A ( G ) = ( a ij ) ∈ R n × n is defined by a ij = (cid:88) e ∈ E sgn e ( v i , v j ) if v i is adjacent to v j , v i is adjacent to v j , then a ij = (cid:88) e ∈ E sgn e ( v i , v j ) = (cid:88) e ∈ E sgn e ( v j , v i ) = a ji . Therefore, A ( G ) is symmetric. he incidence matrix H( G ) = ( η ij ) is the n × m matrix, with entries in {± , } ,defined by η ij = (cid:40) σ ( v i , e j ) if ( v i , e j ) ∈ I , Lemma 2.1 ( [22],Theorem 4.1 ). If G is an oriented hypergraph, then H( G ) T =H( G ∗ ) . The degree matrix of an oriented hypergraph G is defined as D ( G ) := diag( d , d ,. . . , d n ). The Laplacian matrix is defined as L ( G ) := D ( G ) − A ( G ) . The Laplacianmatrix of an oriented hypergraph can also be written in terms of the incidence matrix.
Lemma 2.2 ( [22], Corollary 4.4 ). If G is an oriented hypergraph, then1. L ( G ) = D ( G ) − A ( G ) = H( G )H( G ) T , and2. L ( G ∗ ) = D ( G ∗ ) − A ( G ∗ ) = H( G ) T H( G ) . Since the eigenvalues of a symmetric matrix A ∈ R n × n are real, we will assumethat they are labeled and ordered according to the following convention: λ min ( A ) = λ n ( A ) ≤ λ n − ( A ) ≤ · · · ≤ λ ( A ) ≤ λ ( A ) = λ max ( A ) .
3. Intersection Graphs.
The k -section ( clique graph ) of an oriented hyper-graph G = ( V, E, I , σ ) is the oriented hypergraph [ G ] k with the same vertex set as G and edge set consisting of f ⊆ V that satisfies either of the following:1. | f | = k and f ⊆ e for some e ∈ E , or2. | f | < k and f = e for some e ∈ E .The incidence signs for each f are carried over from e so that σ [ G ] k ( v, f ) = σ G ( v, e ).The k -section of an oriented hypergraph is a generalization of the k -section of ahypergraph (see, for example, Berge [2, p.26]).The strict k -section is the oriented hypergraph (cid:74) G (cid:75) k that is the same as [ G ] k , butwithout condition (2).The intersection graph ( line graph , or representative graph ) of G is the orientedhypergraph Λ( G ) whose vertices are the edges of G , and edges of the form ef whenever e ∩ f (cid:54) = ∅ in G . The incidence signs are carried over to the intersection graph so thatif v ∈ e ∩ f in G , then σ Λ( G ) ( e, ef ) = σ ( v, e ). The intersection graph of an orientedhypergraph simultaneously generalizes the intersection graph of a hypergraph [2, p.31]and the line graph of a signed graph introduced by Zaslavsky [28]. See Figure 3.1 or an example of a 2-section, strict 2-section and intersection graph of an orientedhypergraph G . Λ( G ) J G K [ G ] Fig. 3.1 . The 2-section [ G ] , strict 2-section (cid:74) G (cid:75) and intersection graph Λ( G ) of G fromFigure 2.1. The following theorem shows how to obtain the intersection graph of an orientedhypergraph G by finding the strict 2-section of the dual G ∗ . This generalizes Berge’sknown result for hypergraphs [2, Prop. 1, p.33]. Theorem 3.1. If G is a linear oriented hypergraph, then Λ( G ) = (cid:74) G ∗ (cid:75) .Proof . The equivalences v ∈ V (Λ( G )) ↔ v ∈ E ( G ) ↔ v ∈ V ( G ∗ ) ↔ v ∈ V ( (cid:74) G ∗ (cid:75) ) , verifies V (Λ( G )) = V ( (cid:74) G ∗ (cid:75) ). Similarly, the equivalences e i e j ∈ E (Λ( G )) ↔ ∃ e i , e j ∈ E ( G ) such that ∃ v ∈ V ( G ) such that v ∈ e i ∩ e j , ↔ ∃ e i , e j ∈ V ( G ∗ ) such that ∃ v ∈ E ( G ∗ ) such that e i , e j ∈ v, ↔ ∃ e i , e j ∈ V ( (cid:74) G ∗ (cid:75) ) such that ∃ v ∈ E ( (cid:74) G ∗ (cid:75) ) such that e i , e j ∈ v, ↔ e i e j ∈ E ( (cid:74) G ∗ (cid:75) ) , confirms E (Λ( G )) = E ( (cid:74) G ∗ (cid:75) ). Finally, along with the inherited incidences, theequivalences σ Λ( G ) ( e i , e i e j ) = σ G ( v, e i ) = σ G ∗ ( e i , v ) = σ (cid:74) G ∗ (cid:75) ( e i , e i e j ) , show that the incidence signs are the same for Λ( G ) and (cid:74) G ∗ (cid:75) . Therefore, Λ( G ) = (cid:74) G ∗ (cid:75) . Corollary 3.2. If G is a linear oriented hypergraph, then Λ( G ∗ ) = (cid:74) G (cid:75) . As a generalization of Berge’s result, for any simple oriented signed graph G ,there is a linear oriented hypergraph H that has G as its intersection graph. It turnsout that H = G ∗ is one such oriented hypergraph. Corollary 3.3. If G is a 2-uniform linear oriented hypergraph (that is, G is ansimple oriented signed graph), then Λ( G ∗ ) = G . oreover, for any simple oriented signed graph G , there are infinitely many linearoriented hypergraphs with G as their intersection graph. Corollary 3.4. If G is a 2-uniform linear oriented hypergraph (that is, G is ansimple oriented signed graph), then there exists an infinite family of linear orientedhypergraphs H such that for any H ∈ H , Λ( H ) = G .Proof . Consider the dual G ∗ and pick an edge e ∈ E ( G ∗ ) that has nonzero size i . Let j ∈ { i + 1 , i + 2 , . . . } and let H j be the oriented hypergraph that is identicalto G ∗ except edge e is has an additional j − i new vertices of degree 1 incident to it.These additional vertices also create j − i new incidences, all of which can be givenan orientation of +1 (this choice is arbitrary). Since these new vertices do not createany new edges, or new edges incident to e , it must be that the intersection graphsof G ∗ and H j are the same. By Corollary 3.3, G = Λ( G ∗ ) = Λ( H j ). Therefore, theinfinite family H := { H j : j ≥ i } satisfies the corollary.If k ≥ ∆( G ), the process of enlarging the edge sizes can be continued to obtain a k -uniform oriented hypergraph that has G as its line graph. Theorem 3.5. If G is a 2-uniform linear oriented hypergraph (that is, G is ansimple oriented signed graph), then for all k ≥ ∆( G ) , there exists a k -uniform linearoriented hypergraph H k with Λ( H k ) = G .Proof . Same construction as the previous proof, but enlarge all the edges to havesize k . Since k ≥ ∆( G ) it is guaranteed that all the edges can be enlarged to thedesired size.To illustrate Corollaries 3.3 and 3.4, and Theorem 3.5 see Figure 3.2. G = Λ( G ∗ ) = Λ( H ) G ∗ H Fig. 3.2 . A 2-uniform linear oriented hypergraph G , which is also the line graph of the dual G ∗ and the 5-uniform oriented hypergraph H . Berge shows that every graph is the intersection graph of some linear hypergraph[2, Prop. 2, p.34] as is true for simple signed graphs. In general, a signed graph Σ isthe intersection graph of infinitely many linear oriented hypergraphs. heorem 3.6. If Σ is a simple signed graph, then there is some linear orientedhypergraph H , with Λ( H ) = Σ . Moreover, for all k ≥ ∆(Σ) , there exists a k -uniformlinear oriented hypergraph H k with Λ( H k ) = Σ .Proof . A simple signed graph Σ has many possible orientations β such that G = (Σ , β ) is a 2-uniform linear oriented hypergraph. The result is immediate byTheorem 3.5.For a 2-regular oriented hypergraph G , the intersection graph Λ( G ) and the dual G ∗ are identical. Also, there are infinitely many k -uniform linear oriented hypergraphswhose intersection graphs are G ∗ . Corollary 3.7. If G is a 2-regular linear oriented hypergraph, then Λ( G ) = G ∗ .Moreover, for all k ≥ r ( G ) , there exists a k -uniform oriented hypergraph H k such that Λ( H k ) = G ∗ .Proof . If G is 2-regular, then G ∗ is 2-uniform, hence the results are immediateby Corollaries 3.3 and 3.5.See Figure 3.3 for an example that illustrates Corollary 3.7. G G ∗ = Λ( G ) = Λ( H ) H Fig. 3.3 . A 2-regular linear oriented hypergraph G and its dual G ∗ . The dual G ∗ is the linegraph of both G and the 3-uniform oriented hypergraph H .
4. Matrices of Intersection Graphs.
The strict 2-section (cid:74) G (cid:75) is essentiallythe oriented hypergraph created from the adjacencies in G , so on the level of adjacencymatrices the oriented hypergraphs G and (cid:74) G (cid:75) record the same information. Theorem 4.1. If G is an oriented hypergraph, then A ( G ) = A ( (cid:74) G (cid:75) ) = A ([ G ] ) .Proof . By definition, V ( G ) = V ( (cid:74) G (cid:75) ), so both have adjacency matrices of thesame size. If v i and v j are not adjacenct in G , then they are not adjacent in (cid:74) G (cid:75) and the ( i, j )-entries of A ( G ) and A ( (cid:74) G (cid:75) ) are 0. Otherwise, v i and v j are adjacentand the ( i, j )-entry of A ( G ) is (cid:88) e ∈ E ( G ) sgn e ( v i , v j ) = (cid:88) e ∈ E ( G ) − σ G ( v i , e ) σ G ( v j , e )= (cid:88) f ∈ E ( (cid:74) G (cid:75) ) − σ (cid:74) G (cid:75) ( v i , f ) σ (cid:74) G (cid:75) ( v j , f ) (cid:88) f ∈ E ( (cid:74) G (cid:75) ) sgn f ( v i , v j ) , which is the ( i, j )-entry of A ( (cid:74) G (cid:75) ). This is also the ( i, j )-entry of A ([ G ] ) since anyextra edges of smaller size present in [ G ] would not introduce any new adjacenciesto consider.This adjacency matrix relationship carries over to the dual in a significant way,showing that the dual and intersection graph have the same adjacency matrix. Corollary 4.2. If G is a linear oriented hypergraph, then A ( G ∗ ) = A (Λ( G )) .Proof . The result is an immediate consequence of Theorems 3.1 and 4.1.If G is a k -uniform, we can specialize Lemma 2.2 as follows. Lemma 4.3 ( [22],Corollary 4.5 ). If G is a k -uniform oriented hypergraph, then L ( G ∗ ) = H( G ) T H( G ) = kI − A ( G ∗ ) . Theorem 4.4.
Let G be a k -uniform linear oriented hypergraph. If λ is aneigenvalue of A (Λ( G )) (or A ( G ∗ ) ), then λ ≤ k .Proof . By Corollary 4.2 A (Λ( G )) = A ( G ∗ ), so these matrices can be used inter-changeably. Suppose that x is an eigenvector of A (Λ( G )) with associated eigenvalue λ . By Lemma 4.3 the following simplification can be made: L ( G ∗ ) x = H( G ) T H( G ) x = (cid:0) kI − A (Λ( G ) (cid:1) x = ( k − λ ) x . Hence, k − λ is an eigenvalue of L ( G ∗ ) = H( G ) T H( G ). Since L ( G ∗ ) is positivesemidefinite it must be that k − λ ≥ k ≥ λ . Question 1:
Suppose you are given a signed graph Σ with all adjacency eigen-values satisfying λ ≤ k . Does this mean Σ is the intersection graph of some k -uniformoriented hypergraph? We already know this is always the case if ∆(Σ) ≤ k . For allsigned graphs, it is known that all eigenvalues of A (Σ) satisfy λ ≤ ∆(Σ) [20, Theorem4.3]. So the question remains for situations when all adjacency eigenvalues satisfy λ ≤ k < ∆(Σ) for some integer k . Example 1:
Consider the signed graph Σ in Figure 4.1. In this example λ max ( A (Σ)) ≈ . < k < ∆(Σ) = 5. So is Σ the intersection graph of a 3-uniform or 4-uniform oriented hypergraph? We already know that if k ≥ k -uniform oriented hypergraph H k with Λ( H k ) = Σ by Theorem 3.6. However,it is possible to construct a 3-uniform oriented hypergraph H with Σ as its intersectiongraph, as shown in Figure 4.1. This construction can be modified to the 4-uniform = Λ( H ) H Σ Fig. 4.1 . A signed graph Σ can be oriented so it is also the intersection graph of the depicted3-uniform oriented hypergraph H . A dashed edge denotes a sign of − and a solid edge denotes ofa sign of +1 . The second signed graph Σ is discussed in Example 1. case as well. If, however, we consider the signed graph Σ in Figure 4.1, the situationis different. For this signed graph, λ max ( A (Σ )) ≈ . < k < ∆(Σ ) = 5, but itdoes not seem possible to have a 3 or 4-uniform oriented hypergraph with Σ as itsintersection graph. Is there a nice classification of the exceptional cases?If G is r -regular, the dual relationship of Lemma 2.2 can also be simplified. Lemma 4.5. If G is an r -regular oriented hypergraph, then L ( G ) = H( G )H( G ) T = rI − A ( G ) . Theorem 4.6.
Let G be a r -regular oriented hypergraph. If λ is an eigenvalueof A ( G ) , then λ ≤ r .Proof . Suppose that x is an eigenvector of A ( G ) with associated eigenvalue λ .By Lemma 4.5 L ( G ) x = H( G )H( G ) T x = (cid:0) rI − A ( G ) (cid:1) x = ( r − λ ) x . Hence, r − λ is an eigenvalue of the positive semidefinite matrix L ( G ). Therefore, itmust be that r ≥ λ .An oriented hypergraph and its incidence dual have the same nonzero Laplacianeigenvalues. Lemma 4.7 ( [21],Corollary 4.2 ). If G is an oriented hypergraph, then L ( G ) and L ( G ∗ ) have the same nonzero eigenvalues. Question 2: If G is 2-regular, then Λ( G ) = G ∗ (see Corollary 3.7 above). In thissituation L ( G ), L ( G ∗ ) and L (Λ( G )) all have the same nonzero eigenvalues by Lemma4.7. If G is not 2-regular, when are the eigenvalues of L (Λ( G )) and L ( G ∗ ) different? y Corollary 4.2 the only difference between L (Λ( G )) and L ( G ) in general is: L ( G ∗ ) − L (Λ( G )) = D ( G ∗ ) − D (Λ( G )) . Hence, a sufficient condition can be easily stated, yet a full classification using thestructure of G alone would be more interesting. Proposition 4.8.
Let G be an oriented hypergraph that is not 2-regular.1. If (cid:80) mi =1 d G ∗ i > (cid:80) mi =1 d Λ( G ) i , then ∃ j ∈ { , . . . , m } , with λ j ( L ( G ∗ )) > λ j (cid:0) L (Λ( G )) (cid:1) .2. If (cid:80) mi =1 d G ∗ i < (cid:80) mi =1 d Λ( G ) i , then ∃ j ∈ { , . . . , m } , with λ j ( L ( G ∗ )) < λ j (cid:0) L (Λ( G )) (cid:1) .Proof . The trace of L ( G ∗ ) is tr( L ( G ∗ )) = (cid:80) mi =1 d G ∗ i = (cid:80) mi =1 λ i ( L ( G ∗ )) andthe trace of L (Λ( G )) is tr( L (Λ( G ))) = (cid:80) mi =1 d Λ( G ) i = (cid:80) mi =1 λ i ( L (Λ( G ))). Therefore, astrict increase in trace must result in a strict increase in at least one of the eigenvalues. Example 2:
Consider the oriented hypergraph G in Figure 2.1. The dual G ∗ and intersection graph Λ( G ) are depicted in Figures 2.2 and 3.1. Condition (2) ofProposition 4.8 is met and the conclusion can be seen in the Laplacian eigenvaluesapproximated in Table 4.1. Example 3:
Consider the oriented hypergraph G in Figure 4.2 together withits dual G ∗ and intersection graph Λ( G ). Condition (1) of Proposition 4.8 is metand the conclusion can be seen in the Laplacian eigenvalues approximated in Table4.1. L (Λ( G )) L ( G ∗ ) L (Λ( G )) L ( G ∗ ) λ λ λ λ Table 4.1
Approximate Laplacian eigenvalues of Λ( G ) and G ∗ from Figures 2.2 and 3.1, as well as Λ( G ) and G ∗ in Figure 4.2 . Example 4:
Consider the oriented hypergraph G in Figure 4.2. Here G isnot 2-regular, and yet the eigenvalues of L ( G ∗ ) and L (Λ( G )) are the same. In fact, L ( G ∗ ) = L (Λ( G )). A full classification of when the Laplacian eigenvalues are thesame for G ∗ and Λ( G ) would be of interest. Λ( G ) G ∗ G Λ( G ) G ∗ Fig. 4.2 . Oriented hypergraphs considered in Examples 3 and 4.
5. Switching. A vertex-switching function is any function ζ : V → {− , +1 } . Vertex-switching the oriented hypergraph G = ( H, σ ) means replacing σ with σ ζ ,defined by(5.1) σ ζ ( v, e ) = ζ ( v ) σ ( v, e );producing the oriented hypergraph G ζ = ( H, σ ζ ).An edge-switching function is any function ξ : E → {− , +1 } . Edge-switching theoriented hypergraph G = ( H, σ ) means replacing σ with σ ξ , defined by(5.2) σ ξ ( v, e ) = σ ( v, e ) ξ ( e );producing the oriented hypergraph G ξ = ( H, σ ξ ). To make things more compact wewill write G ( ζ,ξ ) = ( G, σ ( ζ,ξ ) ) when G is both vertex-switched by ζ and edge-switchedby ξ . Switching changes the the adjacency signatures in the following way:sgn ( ζ,ξ ) e ( v i , v j ) = − σ ( ζ,ξ ) ( v i , e ) σ ( ζ,ξ ) ( v j , e ) = − [ ζ ( v i ) σ ( v i , e ) ξ ( e )][ ζ ( v j ) σ ( v j , e ) ξ ( e )]= − ζ ( v i ) σ ( v i , e ) ξ ( e ) σ ( v j , e ) ζ ( v j )= ζ ( v i ) sgn e ( v i , v j ) ζ ( v j ) . The adjacency signatures are conjugated by the vertex-switching ζ and invariant underthe edge-switching ξ . These switching operations can be encoded using matrices.For a vertex-switching function ζ : V → { +1 , − } , we define the diagonal ma-trix D n ( ζ ) := diag (cid:0) ζ ( v ) , ζ ( v ) , . . . , ζ ( v n ) (cid:1) . Similarly for an edge-switching function ξ : E → { +1 , − } , we define D m ( ξ ) := diag (cid:0) ξ ( e ) , ξ ( e ) , . . . , ξ ( e m ) (cid:1) . The followingshows how to calculate the switched oriented hypergraph’s incidence, adjacency andLaplacian matrices extending [22, Propositions 3.1 and 4.3]. Lemma 5.1.
Let G be an oriented hypergraph. Let ζ : V → { +1 , − } be a vertex-switching function on G , and ξ : E → { +1 , − } be an edge-switching function on G .Then . H (cid:0) G ( ζ,ξ ) (cid:1) = D n ( ζ )H( G ) D m ( ξ ) ,2. A (cid:0) G ( ζ,ξ ) (cid:1) = D n ( ζ ) A ( G ) D n ( ζ ) , and3. L (cid:0) G ( ζ,ξ ) (cid:1) = D n ( ζ ) L ( G ) D n ( ζ ) .Moreover,(4) H (cid:0) ( G ∗ ) ( ξ,ζ ) (cid:1) = D m ( ξ )H( G ∗ ) D n ( ζ ) ,(5) A (cid:0) ( G ∗ ) ( ξ,ζ ) (cid:1) = D m ( ξ ) A ( G ∗ ) D m ( ξ ) , and(6) L (cid:0) ( G ∗ ) ( ξ,ζ ) (cid:1) = D m ( ξ ) L ( G ∗ ) D m ( ξ ) . Since switching results in similarity transformations for both the adjacency andLaplacian matrices, it also preserves the respective eigenvalues. A specialized versionof this situation appears in [21, Lemmas 3.1 and 4.1].
Theorem 5.2.
Let G be an oriented hypergraph. Let ζ : V → { +1 , − } be avertex-switching function on G , and ξ : E → { +1 , − } be an edge-switching functionon G . Then1. A ( G ) and A (cid:0) G ( ζ,ξ ) (cid:1) have the same eigenvalues.2. L ( G ) and L (cid:0) G ( ζ,ξ ) (cid:1) have the same eigenvalues.Moreover,(3) A ( G ∗ ) and A (cid:0) ( G ∗ ) ( ξ,ζ ) (cid:1) have the same eigenvalues.(4) L ( G ∗ ) and L (cid:0) ( G ∗ ) ( ξ,ζ ) (cid:1) have the same eigenvalues. Lemma 4.7 can be generalized to find a large family of oriented hypergraphs whichhave the same nonzero Laplacian eigenvalues.
Corollary 5.3.
Let G be an oriented hypergraph. Let ζ and ζ be vertex-switching functions on G , and let ξ and ξ be edge-switching functions on G . Then L (cid:0) G ( ζ ,ξ ) (cid:1) and L (cid:0) ( G ∗ ) ( ξ ,ζ ) (cid:1) have the same nonzero eigenvalues. If two oriented hypergraphs are the same up to a vertex or edge switching, then thecorresponding duals, strict 2-sections and intersection graphs have related switchingrelationships.
Theorem 5.4.
Let G = ( H, σ ) and G = ( H, σ ) be linear oriented hyper-graphs. Let ζ be a vertex-switching function on G and G , and ξ be an edge-switchingfunction on G and G . If G = G ( ζ,ξ )2 , then1. G ∗ = ( G ∗ ) ( ξ,ζ ) .Moreover, there exists edge switching functions ˆ ξ on (cid:74) G (cid:75) and ˆ ζ on Λ( G ) such that(2) (cid:74) G (cid:75) = (cid:74) G (cid:75) ( ζ, ˆ ξ )2 , and(3) Λ( G ) = Λ( G ) ( ξ, ˆ ζ ) . roof . Since ζ is a vertex-switching function on G and G , by duality, ζ isan edge-switching function on G ∗ and G ∗ . Similarly ξ becomes a vertex-switchingfunction on G ∗ and G ∗ . Now (1) follows from the incidence simplifications: σ ( ξ,ζ ) G ∗ ( e, v ) = ξ ( e ) σ G ∗ ( e, v ) ζ ( v ) = ζ ( v ) σ G ( v, e ) ξ ( e ) = σ ( ζ,ξ ) G ( v, e ) = σ G ( v, e )= σ G ∗ ( e, v ) . The strict 2-sections (cid:74) G (cid:75) and (cid:74) G (cid:75) both have the same vertex set as H and therefore ζ is also a vertex-switching function for both oriented hypergraphs. Define the edge-switching function ˆ ξ : E ( (cid:74) G (cid:75) ) → { +1 , − } as an extension of ξ by the following rule:if f ∈ E ( (cid:74) G (cid:75) ) is derived from e ∈ E ( G ) by definition (see the beginning of Section3), then ˆ ξ ( f ) = ξ ( e ). Now (2) follows from the following incidence calculation: σ ( ζ, ˆ ξ ) (cid:74) G (cid:75) ( v, f ) = ζ ( v ) σ (cid:74) G (cid:75) ( v, f ) ˆ ξ ( f ) = ζ ( v ) σ (cid:74) G (cid:75) ( v, f ) ξ ( e ) = ζ ( v ) σ G ( v, e ) ξ ( e )= σ ( ζ,ξ ) G ( v, e )= σ G ( v, e )= σ (cid:74) G (cid:75) ( v, f ) . Finally, the result of (3) follows from (1), (2) and Theorem 3.1.
6. Balanced Incomplete Block Designs.
The matrix relationships found inthe above sections produce a connection to balanced incomplete block designs. Thereis a similar standard matrix relationship that appears in design theory which can bederived from a specialized oriented hypergraph. The definitions and design theoryresults below are taken directly from [7].Suppose G = ( H, +1) represents an oriented hypergraph G with underlying hy-pergraph H and all incidences labeled +1. Theorem 6.1.
Suppose G = ( H, +1) is a k -uniform, r -regular oriented hyper-graph where any two distinct vertices are adjacent exactly λ times. L ( G ) = H( G )H( G ) T = ( r − λ ) I + λJ. Proof . Since G is r -regular, L ( G ) = H( G )H( G ) T = rI − A ( G ), by Lemma 4.5.Since all incidences are labeled +1, this forces all adjacency signs to be − v i and v j are adjacent exactly λ times, if i (cid:54) = j , then the ( i, j )-entry of A ( G ) is (cid:80) e ∈ E sgn e ( v i , v j ) = − λ . Other-wise, a ii = 0. Hence, the result follows by further simplifying the initial equation: rI − A ( G ) = rI − ( λI − λJ ). balanced incomplete block design (BIBD) is a pair ( V, B ), where V is a v -set(i.e., v = | V | ) and B is a collection of b k -subsets of V (called blocks ) such that eachelement of V is contained in exactly r blocks, and any 2-subset of V is contained inexactly λ blocks. The numbers v , b , r , k and λ are called the parameters of the BIBD.The incidence matrix of a BIBD ( V, B ) with parameters v , b , r , k and λ is a v × b matrix C = ( c ij ), where c ij = 1 when the i th element of V occurs in the j th block of B , and c ij = 0, otherwise.A BIBD ( V, B ) can be thought of as a k -uniform, r -regular oriented hypergraph G = ( H, +1) with V ( H ) = V , E ( H ) = B and for any two distinct vertices v i and v j are adjacent exactly λ times. From the Theorem 6.1, the same result known forBIBD can be established. Corollary 6.2 ( [7],Theorem 1.8 ). Suppose C is the incidence matrix of abalanced incomplete block design (BIBD) with parameters v , b , r , k , and λ . Then CC T = ( r − λ ) I + λJ. REFERENCES[1] B. D. Acharya, Signed intersection graphs, J. Discrete Math. Sci. Cryptogr. 13 (6) (2010)553–569.[2] C. Berge, Hypergraphs, vol. 45 of North-Holland Mathematical Library, North-Holland Pub-lishing Co., Amsterdam, 1989, combinatorics of finite sets, Translated from the French.[3] J.-C. Bermond, F. Ergincan, M. Syska, Line directed hypergraphs, in: D. Naccache (ed.),Cryptography and Security: From Theory to Applications, vol. 6805 of Lecture Notes inComputer Science, Springer Berlin Heidelberg, 2012, pp. 25–34.[4] A. Bj¨orner, M. Las Vergnas, B. Sturmfels, N. White, G. M. Ziegler, Oriented Matroids, vol. 46of Encyclopedia of Mathematics and its Applications, 2nd ed., Cambridge University Press,Cambridge, 1999.[5] Vinciane Chen, Angeline Rao, Lucas J. Rusnak, and Alex Yang,
A characterization of ori-ented hypergraphic balance via signed weak walks , Linear Algebra and its Applications485