On Computing the Number of Short Cycles in Bipartite Graphs Using the Spectrum of the Directed Edge Matrix
aa r X i v : . [ c s . I T ] M a r On Computing the Number of Short Cycles inBipartite Graphs Using the Spectrum of theDirected Edge Matrix
Ali Dehghan, and Amir H. Banihashemi,
Senior Member, IEEE
Abstract
Counting short cycles in bipartite graphs is a fundamental problem of interest in many fields in-cluding the analysis and design of low-density parity-check (LDPC) codes. There are two computationalapproaches to count short cycles (with length smaller than g , where g is the girth of the graph) inbipartite graphs. The first approach is applicable to a general (irregular) bipartite graph, and uses thespectrum { η i } of the directed edge matrix of the graph to compute the multiplicity N k of k -cycles with k < g through the simple equation N k = P i η ki / (2 k ) . This approach has a computational complexity O ( | E | ) , where | E | is number of edges in the graph. The second approach is only applicable to bi-regular bipartite graphs, and uses the spectrum { λ i } of the adjacency matrix (graph spectrum) and thedegree sequences of the graph to compute N k . The complexity of this approach is O ( | V | ) , where | V | is number of nodes in the graph. This complexity is less than that of the first approach, but the equationsinvolved in the computations of the second approach are very tedious, particularly for k ≥ g + 6 . Inthis paper, we establish an analytical relationship between the two spectra { η i } and { λ i } for bi-regularbipartite graphs. Through this relationship, the former spectrum can be derived from the latter throughsimple equations. This allows the computation of N k using N k = P i η ki / (2 k ) but with a complexityof O ( | V | ) rather than O ( | E | ) . Index Terms:
Counting cycles, short cycles, bipartite graphs, Tanner graphs, low-density parity-check(LDPC) codes, bi-regular bipartite graphs, irregular bipartite graphs, directed edge matrix, girth. I. INTRODUCTION
Bipartite graphs appear in many fields of science and engineering to represent systems thatare described by local constraints on different subsets of variables involved in the descriptionof the system. In such a representation, the nodes on one side of the bipartition represent thevariables while the nodes on the other side are representative of the constraints. One example is the Tanner graph representation of low-density parity-check (LDPC) codes, where variablenodes represent the code bits and the constraints are parity-check equations. In the bipartitegraph representation of systems, the cycle distribution of the graph often plays an important rolein understanding the properties of the system. For example, the performance of LDPC codes,both in waterfall and error floor regions, is highly dependent on the distribution of short cyclesof the Tanner graph [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].Motivated by this, in the coding community, there has been a large body of work on thedistribution and counting of cycles in bipartite graphs, see, e.g., [3], [11], [12], [13], [14].Generally, counting cycles of a given length in a given graph is known to be NP-hard [15]. Theproblem remains NP-hard even for the family of bipartite graphs [16]. There are, in general,two computational approaches to count the number of short cycles in bipartite graphs. Thefirst approach is applicable to any (irregular) bipartite graph, and is described in the followingtheorem.
Theorem 1. [11] Consider a bipartite graph G with the directed edge matrix A e , and let { η i } be the spectrum of A e . Then, the number of k -cycles in G is given by N k = P i η ki k , for k < g ,where g is the girth of G . The result of Theorem 1 follows from the property of A e that the number of tailless back-trackless closed (TBC) walks of length k in G is equal to tr ( A ke ) / k , where tr ( A e ) denotesthe trace of A e . This together with the fact that the set of TBC walks of length less than g coincides with the set of cycles of the same size [12] prove the result. To use Theorem 1, oneneeds to calculate the eigenvalues of A e . This has a complexity of O ( | E | ) , where | E | is numberof edges in the graph [17].The second approach, which was introduced by Blake and Lin [14] and extended by Dehghanand Banihashemi [18], uses the spectrum of the adjacency matrix and the degree distribution ofthe graph. It has a lower complexity of O ( | V | ) , where | V | is number of nodes in the graph,but is only applicable to bi-regular bipartite graphs. One drawback of this approach is that therecursive equations for calculating N k are tedious, particularly for values of k ≥ g + 6 . Thefollowing theorem describes the general calculation of N i , for any g ≤ i ≤ g − , and thespecifics of the calculation of N g +4 . Theorem 2. [18] For a given ( d v , d c ) -regular bipartite graph G , the number of i -cycles, g ≤ i ≤ g − , is given by N i = [ | V | X j =1 λ ij − Ω i ( d v , d c , G ) − Ψ i ( d v , d c , G )] / (2 i ) , (1) where { λ j } | V | j =1 is the spectrum of G , and Ω i ( d v , d c , G ) and Ψ i ( d v , d c , G ) are the number ofclosed cycle-free walks of length i and closed walks with cycle of length i in G , respectively.For i = g + 4 , we have Ψ g +4 ( d v , d c , G )2( g + 4) = N g +2 × [ g + 22 ( d v + d c ) − ( g + 2)]+ N g × [ g d v − d c −
1) + g d c − d v − N g × (cid:16) [ (cid:18) g (cid:19) + g d v − + [ (cid:18) g (cid:19) + g d c − + ( g ( d v − d c − (cid:17) + N g × (cid:16)(cid:18) g (cid:19) + 2 g + ( g + 2) × ( g d v −
2) + g d c − (cid:17) , and Ω g +4 ( d v , d c , G ) = n × S d v ,d c ,g +4 + m × S d c ,d v ,g +4 , (2) where n and m are the number of variable and check nodes in G , respectively, and S d v ,d c ,g +4 ( S d c ,d v ,g +4 ) represents the number of closed cycle-free walks of length g + 4 from a variablenode v (a check node c ) to itself. (Generating functions are used to compute functions S x,y,i recursively [14].) In this work, we investigate the relationship between the above two approaches. In particu-lar, our goal is to find the relationship between the two spectra { η i } and { λ i } for bi-regularbipartite graphs. We show that the former spectrum includes eigenvalues ± , ± p − ( d v − ,and ± p − ( d c − . The remaining eigenvalues of A e are related to the graph spectrum { λ i } through simple quadratic equations whose coefficients are determined by the node degrees d v and d c . This allows one to compute N k using Theorem 1, but through the calculation of thegraph spectrum { λ i } rather than the direct calculation of { η i } . As a result, the computationalcomplexity reduces to O ( | V | ) rather than O ( | E | ) , while avoiding the tedious equations ofTheorem 2.The organization of the rest of the paper is as follows: In Section II, we present some definitionsand notations. Section III contains our result on the relationship between the two spectra { λ i } and { η i } , and the derivation of the latter from the former. The paper is concluded in Section IV. II. D
EFINITIONS AND NOTATIONS
A graph G = ( V, E ) is a set V ( G ) of nodes and a multiset E ( G ) of unordered pairs of nodes,called edges. If { v, u } ∈ E , we say that there is an edge between v and u (i.e., v and u areadjacent). We may also use notations uv or vu for the edge { v, u } . We say that a graph G issimple, if it does not have any loop (i.e., no edge of the form { v, v } ) or parallel edges (i.e.,no two edges between the two same nodes). A directed graph (digraph) D = ( V, E ) is a set V of nodes and a multiset E of ordered pairs of nodes called arcs. For an arc e = ( u, w ) , wedefine the origin of e to be o ( e ) = u , and the terminus of e to be t ( e ) = w . The inverse arc of e , denoted by e , is the arc formed by switching the origin and terminus of e . A digraph D iscalled symmetric if whenever ( u, w ) is an arc of D , its inverse arc ( w, u ) is as well. For eachgraph G , its symmetric digraph D ( G ) is defined by replacing each edge of G with two arcs inopposite directions. See Fig. 1. Thus, there is a simple correspondence between G and D ( G ) . v u v u v u u v Fig. 1. A graph G and its symmetric digraph D ( G ) . In a graph G , the number of edges incident to a node v is called the degree of v , and isdenoted by d ( v ) . Also, ∆( G ) and δ ( G ) are used to denote the maximum and minimum degreeof G . For every node v ∈ V ( G ) , the set N ( v ) denotes the set of neighbors of v in G .For a graph G , a walk of length c is a sequence of nodes v , v , . . . , v c +1 in V such that { v i , v i +1 } ∈ E , for all i ∈ { , . . . , c } . A walk can alternatively be represented by its sequenceof edges. A walk v , v , . . . , v k +1 is a path if all the nodes v , v , . . . , v k are distinct. A walkis called a closed walk if the two end nodes are the same, i.e., if v = v k +1 . Under the samecondition, a path is called a cycle . We denote cycles of length k , also referred to as k -cycles,by C k . We use N k for | C k | . The length of the shortest cycle(s) in a graph is called girth and isdenoted by g .Consider a walk W of length k represented by the sequence of edges e i , e i , . . . , e i k . Thewalk W is backtrackless if e i s = e i s +1 , for any s ∈ { , . . . , k − } . Also, the walk W is tailless if e i = e i k . In this paper, we use the term TBC walk to refer to a tailless backtrackless closedwalk.A graph G is connected , if there is a path between any two nodes of G . A graph G = ( V, E ) is called bipartite , if the node set V can be partitioned into two disjoint subsets U and W ,i.e., V = U ∪ W and U ∩ W = ∅ , such that every edge in E connects a node from U to anode from W . A graph is bipartite if and only if the lengths of all its cycles are even. Tannergraphs of LDPC codes are bipartite graphs, in which U and W are referred to as variable nodes and check nodes , respectively. Parameters n and m in this case are used to denote | U | and | W | ,respectively. Parameter n is the code’s block length and the code rate R satisfies R ≥ − ( m/n ) .A bipartite graph G = ( U ∪ W, E ) is called bi-regular , if all the nodes on the same side of thebipartition have the same degree, i.e., if all the nodes in U have the same degree d u and all thenodes in W have the same degree d w . In the rest of the paper, we sometimes use notations d v and d c as a replacement for d u and d w , respectively, to follow the notations commonly used incoding to denote variable and check node degrees, respectively. It is clear that, for a bi-regulargraph, | U | d u = | W | d w = | E ( G ) | . A bipartite graph that is not bi-regular is called irregular . Abipartite graph G ( U ∪ W, E ) is called complete , and is denoted by K | U | , | W | , if every node in U is connected to every node in W . The degree sequences of a bipartite graph G are defined asthe two monotonic non-increasing sequences of the node degrees on the two sides of the graph.For instance, the complete bipartite graph K , has degree sequences (4 , , and (3 , , , .The adjacency matrix of a graph G is a | V | × | V | matrix A = [ a ij ] , where a ij is the number ofedges connecting the node i to the node j , for all i, j ∈ V . Similarly, The adjacency matrix of adigraph D is the matrix A D = [ b ij ] , where b ij is one if and only if ( i, j ) ∈ E ( D ) . The adjacencymatrix A is symmetric, and since we assumed that G has no parallel edges, then a ij ∈ { , } ,for all i, j ∈ V . Moreover, since G has no loops, then a ii = 0 , for all i ∈ V .An eigenvalue of A is a number λ such that A −→ v = λ −→ v , for some nonzero vector −→ v .(Throughout the paper all vectors are assumed to be column vectors.) The vector −→ v is thencalled an eigenvector of A . The set of the eigenvalues { λ i } of the adjacency matrix A of a graph G is called the spectrum of G . The determinant det( λI − A ) , where I is the identity matrix,is called the characteristic polynomial of A (with variable λ ). The roots of this polynomialare the eigenvalues of A . An eigenvalue λ ′ of A is said to have multiplicity i if, when thecharacteristic polynomial is factorized into linear factors, the factor ( λ − λ ′ ) appears i times. If λ is an eigenvalue of A , then the subspace {−→ v : A −→ v = λ −→ v } is called the eigenspace of A associated with λ . The dimension of this eigensapce is at most the multiplicity of λ .There are some known results about the eigenvalues and eigenvectors of the adjacency matrix A that we review below and use them in our work (see, e.g., [19]). (1) If λ is an eigenvalueof A , then λ is an eigenvalue of A . (2) [Perron-Frobenius, Symmetric Case] Let A be theadjacency matrix of a connected graph G , and let λ ≥ λ ≥ . . . ≥ λ | V | be the spectrum of G . Then, λ > λ (i.e., the multiplicity of the largest eigenvalue of A is one). (3) The largesteigenvalue of bi-regular bipartite graphs is √ d v d c [20]. (4) A graph is bipartite if and only ifits spectrum is symmetric about the origin. (5) By Properties (2) and (4), in connected bipartitegraphs, the multiplicity of the smallest eigenvalue is also one. (6) By Property (4), for a givenbipartite graph G , if λ i is an eigenvalue of A with multiplicity m i , then − λ i is also an eigenvaluewith multiplicity m i . Thus, the spectrum of A has the following form {± λ m , . . . , ± λ m r r } , forsome r ≥ , and we have P ri =1 × m i = | V | . (7) The adjacency matrix A of G has | V ( G ) | linearly independent eigenvectors, such that for each ≤ i ≤ r , there are m i linearly independenteigenvectors associated with each eigenvalue λ i and − λ i .Another important property of the adjacency matrix is that the number of walks between anytwo nodes of the graph can be determined using the powers of this matrix. In other words, theentry in the i th row and the j th column of A k , [ A k ] ij , is the number of walks of length k betweennodes i and j . Consequently, the total number of closed walks of length k in G is tr ( A k ) , where tr ( · ) is the trace of a matrix. It is well-known that tr ( A k ) = P | V | i =1 λ ki , and thus the multiplicityof closed walks of different length in a graph can be obtained using the spectrum of the graph.For a given graph G , the directed edge matrix A e , is a | E | × | E | matrix defined as follows.For each edge e i = { v, u } in G , we consider two opposite arcs ( v, u ) , ( u, v ) , and denote themby f i and f | E ( G ) | + i (i.e., f i = f | E ( G ) | + i ). We then define ( A e ) i,j = , if t ( f i ) = o ( f j ) and f i = f j , otherwise . (3)In other words, for a given graph G , we consider its associated symmetric digraph D ( G ) , andthen calculate A e from D ( G ) using (3). For example, for graphs G and D ( G ) in Fig. 1, we have A e = .The number of k -cycles, g ≤ k ≤ g − , in a bipartite graph G can be obtained from thespectrum { η i } of A e using Theorem 1.The rank of a matrix B , denoted by Rank ( B ) , is the dimension of the vector space generatedby its columns. This corresponds to the maximum number of linearly independent columns of A . The rank is also the dimension of the space spanned by the rows of B . Thus, if B is an m × n matrix, then Rank ( B ) = Rank ( B t ) ≤ min { m, n } , (4)where B t is the transpose of B . The kernel (null space) of a matrix B is the set of solutionsto the equation B −→ x = −→ , where −→ is the zero vector. The dimension of the null space of B is called the nullity of B and is denoted by N ull ( B ) . For an m × n matrix B , we have(Rank-Nullity Theorem): Rank ( B ) + N ull ( B ) = n . (5)III. T HE R ELATIONSHIP BETWEEN THE S PECTRA OF A e AND A FOR B I - REGULAR B IPARTITE G RAPHS , AND THE N EW M ETHOD TO C OUNT S HORT C YCLES
In [21], it was shown that for a regular graph G , the eigenvalues of A e can be computedfrom those of A . A key component in the derivations of [21] is the special properties that A e has as a result of the regularity of the graph. For the bi-regular graphs, considered in this work,however, such properties do not exist and thus the derivations are much different. In this section,we derive the spectrum { η i } of A e from the graph spectrum { λ i } for bi-regular bipartite graphs,and then use the results to count the short cycles of the graph by Theorem 1.To derive our results, we first define an auxiliary matrix e A as a function of A . We then findthe eigenvalues { ξ i } of e A , which are on the one hand related to { λ i } , and on the other hand to { η i } . Through these relationships, we derive { η i } from { λ i } . In the following, for simplicity,we use notations q and q to denote d v − and d c − , respectively.For a bi-regular bipartite graph G = ( U ∪ W, E ) , let e A = [ e a ( u,w ) , ( x,y ) ] u,w,x,y ∈ V ( G ) be a | V ( G ) | ×| V ( G ) | matrix such that the entries of e A are given by e a ( u,w ) , ( x,y ) = a uw a xy δ wx (1 − δ uy ) , (6)where δ uw is the Kronecker delta (which is equal to if u = w , and equal to zero, otherwise),and a uw is the ( u, w ) th entry of the adjacency matrix A of G . In the rest of the paper, we assumethat the rows and columns of e A are sorted in the following order: First, the set { ( u, w ) : u ∈ U, w ∈ W, uw ∈ E ( G ) } , second { ( w, u ) : u ∈ U, w ∈ W, uw ∈ E ( G ) } , and finally, other pairs { ( u, w ) , ( w, u ) : u ∈ U, w ∈ W, uw / ∈ E ( G ) } . Note that the union of the first two sets is the setof directed edges in the symmetric digraph D ( G ) associated with G . Also, by (6), e a ( u,w ) , ( x,y ) = 1 if and only if we have(i) a uw a xy = 1 (i.e., f i = ( u, w ) , f j = ( x, y ) ∈ E ( D ( G )) ),(ii) δ wx = 1 (i.e., t ( f i ) = o ( f j ) ), and(iii) (1 − δ uy ) = 1 (i.e., f i = f j ) ).Thus, by (3), the matrix e A has the following form e A = A e (2 | E | ) × ( | V | − | E | ) ( | V | − | E | ) × (2 | E | ) ( | V | − | E | ) × ( | V | − | E | ) , (7)and by (7), we have the following result. Lemma 1.
The eigenvalues of e A are the same as those of A e with the addition of | V | − | E | zero eigenvalues. Furthermore, since G is bipartite, and based on the labeling of rows and columns (i.e., first,are listed pairs { ( u, w ) : u ∈ U, w ∈ W, uw ∈ E ( G ) } , followed by pairs { ( w, u ) : u ∈ U, w ∈ W, uw ∈ E ( G ) } ), A e has the following form A e = | E |×| E | B e C e | E |×| E | , (8)where B e and C e are | E | × | E | matrices. As an example, by the ordering just described( ( u , v ) , ( u , v ) , ( u , v ) , ( u , v ) are the first arcs, followed by their inverse arcs in the sameorder), for the graph G shown in Fig. 1, we have A e = . From (7) and (8), one can see that the matrix e A has the following form: e A = B e C e | E |×| E | | E |×| E | C e B e (2 | E | ) × ( | V | − | E | ) ( | V | − | E | ) × (2 | E | ) ( | V | − | E | ) × ( | V | − | E | ) , (9)or equivalently, e A = A e (2 | E | ) × ( | V | − | E | ) ( | V | − | E | ) × (2 | E | ) ( | V | − | E | ) × ( | V | − | E | ) (10)It is easy to see that (( u, w ) , ( x, y )) th entry of the element of e A (denoted by e a u,w ) , ( x,y ) ) is givenby e a u,w ) , ( x,y ) = , if uw, wx, xy ∈ E, x = u, y = w , otherwise . (11)We thus have e a u,w ) , ( x,y ) = a uw a wx a xy (1 − δ xu )(1 − δ yw ) . (12)Next, we study the structure of eigenvectors of e A . Lemma 2.
Consider a number ξ = 0 and a vector −→ φ of size | V | , and denote the elementthat corresponds to the pair ( x, y ) in the vector −→ φ by φ ( x,y ) . Then, −→ φ is an eigenvector of e A associated with eigenvalue ξ if and only if, for each pair ( u, w ) , where u ∈ U and w ∈ W , wehave ξφ ( u,w ) = a uw X x ∈ U a wx X y ∈ W a xy φ ( x,y ) − a uw X x ∈ U a wx φ ( x,w ) − a uw X y ∈ W a uy φ ( u,y ) + a uw φ ( u,w ) , (13) and for each pair ( w, u ) , where w ∈ W and u ∈ U , we have ξφ ( w,u ) = a wu X y ∈ W a uy X x ∈ U a yx φ ( y,x ) − a wu X y ∈ W a uy φ ( y,u ) − a wu X x ∈ U a wx φ ( w,x ) + a wu φ ( w,u ) , (14) and for all the other pairs ( x, y ) , φ ( x,y ) = 0 .Proof. By the definition of eigenvalue/eigenvector and (12), it is clear that for ξ = 0 , we musthave φ ( x,y ) = 0 , for all cases where nodes x and y are on the same side of the graph. On the otherhand, for each pair ( u, w ) , where u ∈ U and w ∈ W , by the definition of eigenvalue/eigenvectorand (12), we have: ξφ ( u,w ) = X x ∈ U X y ∈ W a uw a wx a xy (1 − δ xu )(1 − δ yw ) φ ( x,y ) = a uw X x ∈ U a wx X y ∈ W a xy (1 − δ xu )(1 − δ yw ) φ ( x,y ) = a uw X x ∈ U a wx X y ∈ W a xy φ ( x,y ) − a uw X x ∈ U a wx φ ( x,w ) − a uw X y ∈ W a uy φ ( u,y ) + a uw φ ( u,w ) Equation (14) is derived similarly.
A. From the non-zero eigenvalues of A to the eigenvalues of e A Lemma 3.
Let λ = 0 be an eigenvalue of the adjacency matrix A . Then the solutions of thequadratic equation ξ + ( − λ + q + q ) ξ + q q = 0 are two eigenvalues of e A .Proof. Let λ be an eigenvalue of the adjacency matrix A with a corresponding eigenvector −→ µ = [ µ u , . . . , µ u n , µ w , . . . , µ w m ] t (note that the elements of the eigenvector are sorted bylisting the elements corresponding to the nodes in U first, followed by those corresponding tothe nodes in W ). By using −→ µ , we define a vector −→ φ of size | V | in the following way (theelement corresponding to the pair ( x, y ) , x ∈ V, y ∈ V , in −→ φ is denoted by φ ( x,y ) ): φ ( x,y ) = a xy ( µ y − f µ x ) , if x ∈ U, y ∈ W,a xy ( µ y − f µ x ) , if x ∈ W, y ∈ U, , otherwise , (15)where f and f are constant numbers. Now, we show that by the proper choice of f and f ,the vector −→ φ is an eigenvector of e A , and in the process find the corresponding eigenvalues ξ . By substituting (15) in (13), we have: ξφ ( u,w ) = a uw X x ∈ U a wx X y ∈ W a xy ( µ y − f µ x ) − a uw X x ∈ U a wx ( µ w − f µ x ) − a uw X y ∈ W a uy ( µ y − f µ u )+ a uw ( µ w − f µ u )= a uw X x ∈ U a wx (cid:16) λµ x − ( q + 1) f µ x (cid:17) − a uw µ w ( q + 1) + a uw f λµ w − a uw λµ u + a uw f µ u ( q + 1)+ a uw µ w − a uw f µ u = a uw λ µ w − a uw ( q + 1) f λµ w − a uw µ w ( q + 1) + a uw f λµ w − a uw λµ u + a uw f µ u ( q + 1)+ a uw µ w − a uw f µ u = a uw µ w (cid:16) λ − λf q − q (cid:17) − a uw µ u (cid:16) λ − f q (cid:17) , (16)where in the second and third last steps, we have used the definition of eigenvalue/eigenvectorof A . From (16), and considering ξ = 0 , we have: ξφ ( u,w ) = a uw ξ (cid:16) λ − λf q − q ξ µ w − λ − f q ξ µ u (cid:17) . (17)From (17) and (15), we obtain: λ − λf q − q ξ = 1 λ − f q ξ = f (18)By solving (18), we have (note that since λ = 0 , by (18), we have ξ = − q ): f = λξ + q , (19) and ξ + ( − λ + q + q ) ξ + q q = 0 . (20)Similarly, by substituting (15) in (14), and taking the same steps as those taken in the derivationof (16), we have: ξφ ( w,u ) = a wu ξ (cid:16) λ − λf q − q ξ µ u − λ − f q ξ µ w (cid:17) . (21)From (21) and (15), we have: λ − λf q − q ξ = 1 λ − f q ξ = f . (22)By solving (22), we obtain (since λ = 0 , by (22), ξ = − q ): f = λξ + q , (23)and the same equation as in (20).Therefore, by solving (20), we find the eigenvalues ξ of e A corresponding to λ , and then bysubstituting the obtained ξ in (19) and (23), we find the constants f and f . These are thenreplaced in (15) to obtain the corresponding eigenvectors of e A .Next, we discuss how the eigenvalues of A e can be computed from those of e A . B. From the spectrum of e A to that of A e Lemma 4. [11] Let G be a bi-regular bipartite graph and A e be its directed edge matrix. Then,the eigenvalues of A e are symmetric with respect to the origin. Moreover, η is an eigenvalueof A e if and only if ± η are eigenvalues of A e . Lemma 5.
Let G be a bi-regular bipartite graph. Then the spectrum of e A can be computed fromthat of e A , i.e., if e A has an eigenvalue ξ with multiplicity m , then e A has eigenvalues ±√ ξ ,each with multiplicity m/ .Proof. The proof follows from Lemma 4, (7) and (10).Using Lemmas 1 and 5, one can obtain the spectrum of A e from that of e A . C. From the spectrum of A to that of A e Theorem 3.
Let G = ( V = U ∪ W, E ) be a connected bi-regular bipartite graph such that eachnode in U has degree q + 1 and each node in W has degree q + 1 , where q ≥ , q ≥ and q ≥ q . Also, assume that | U | = n and | W | = m . The eigenvalues of the directed edge matrix A e of G can then be computed from the eigenvalues of the adjacency matrix A as follows: Step 1.
For each strictly negative eigenvalue λ of A , use Equation (20) to find two solutions. Foreach solution ξ = 1 , the numbers ±√ ξ are eigenvalues of A e , each with the same multiplicityas that of λ in the spectrum of A . (The total number of eigenvalues of A e obtained in this stepis m + n ) − N ull ( A ) − .) Step 2.
Matrix A e also has the eigenvalues ±√− q and ±√− q . The multiplicity of each ofthe eigenvalues ±√− q ( ±√− q ) is n − Rank ( A ) / ( m − Rank ( A ) / ). (The total number ofof eigenvalues of A e obtained in this step is m + n ) − Rank ( A ) = 2 N ull ( A ) .) Step 3.
Furthermore, Matrix A e has eigenvalues ± , each with multiplicity | E | − ( m + n ) + 1 .(The total number of eigenvalues in this step is | E | − m + n ) + 2 .)Proof. In the following, we find the set of eigenvalues of e A and their multiplicities, and thenuse Lemmas 1 and 5 to obtain the set of eigenvalues of A e .Suppose that the spectrum of A is {± λ m , . . . , ± λ m r r } , for some r ≥ , where P ri =1 × m i = | V | . For each i , ≤ i ≤ r , there are m i linearly independent eigenvectors −→ µ i, , . . . , −→ µ i,m i ,associated with the eigenvalue λ i .For each i , let ξ i and ξ i be the two eigenvalues obtained from (20) by replacing λ by λ i (note that the solutions of (20) for λ = − λ i are the same as those for λ = λ i ). We considerthree cases that cover all possible scenarios. Case A: λ i = 0 and ξ i = 1 ; Case B: λ i = 0 ; andCase C: λ i = 0 and ξ i = 1 . (Cases A, B and C correspond to Steps 1, 2 and 3 of the derivationof all the eigenvalues of A e . Note that, for each of Cases A, B and C, in the following, we finda lower bound on the multiplicity of the eigenvalues of A e (or those of e A ) that are obtained inthose cases. Based on the fact that the sum of the obtained lower bounds is equal to | E | ( | V | )for A e ( e A ), we conclude that in each case, the multiplicity of the eigenvalues is exactly equalto the lower bound.) Note that ±√− q and ±√− q are solutions of (20) for λ = 0 . Case A. ( λ i = 0 and ξ i = 1 ) In this case, we show that for each i , the multiplicity of ξ i is atleast × m i . Consider vectors −→ φ i, , . . . , −→ φ i,m i , each of size | V | , corresponding to eigenvectors −→ µ i, , . . . , −→ µ i,m i of A associated with eigenvalue λ i , respectively. Assume that the element ( x, y ) , x ∈ V, y ∈ V ,of each vector −→ φ i,j is derived from the elements of the corresponding vector −→ µ i,j using thefollowing equation: φ ( x,y ) = a xy ( µ y − f µ x ) , if x ∈ U, y ∈ W, , otherwise , (24)where f = λ i ξ i + q . Using simple calculations, one can see that for each j , we have e A −→ φ i,j = ξ i −→ φ i,j , and thus, −→ φ i, , . . . , −→ φ i,m i are eigenvectors associated with the eigenvalue ξ i .Also, consider vectors −→ ρ i, , . . . , −→ ρ i,m i , each of size | V | , corresponding to eigenvectors −→ µ i, , . . . , −→ µ i,m i of A associated with eigenvalue λ i , respectively. Assume that the element ( x, y ) , x ∈ V, y ∈ V , of each vector −→ ρ i,j is derived from the elements of the correspondingvector −→ µ i,j using the following equation: ρ ( x,y ) = a xy ( µ y − f µ x ) , if x ∈ W, y ∈ U, , otherwise , (25)where f = λ i ξ i + q . For each j , we have e A −→ ρ i,j = ξ i −→ ρ i,j , and thus, vectors −→ ρ i, , . . . , −→ ρ i,m i are also eigenvectors associated with the eigenvalue ξ i .Regarding the dependency within each of the two groups of eigenvectors {−→ φ i,j } and {−→ ρ i,j } ,we have the following fact whose proof is provided in Appendix V-A. Fact 1.
The vectors −→ φ i, , . . . , −→ φ i,m i are linearly independent. So are the vectors −→ ρ i, , . . . , −→ ρ i,m i .Fact 1 together with the fact that there is no overlap between the location of non-zero elementsin any vector in the set {−→ φ i,j } and that of any vector in the set {−→ ρ i,j } prove that the multiplicityof ξ i , in Case A, is at least × m i .By Lemmas 1 and 5, the number of eigenvalues η of A e that are obtained from Case A is thesame as the number of eigenvalues ξ of e A that are obtained for this case. To count the totalnumber of eigenvalues ξ of e A , we note that the total number of non-zero eigenvalues λ of A is m + n − N ull ( A ) , out of which half are negative. This together with the fact that each eigenvalue As explained before, this lower bound is tight. λ results in two eigenvalues ξ and that if the multiplicity of λ is m , then the multiplicity ofeach resulting ξ is m implies that the total number of eigenvalues ξ is m + n − N ull ( A )) .For Case A, however, we have excluded ξ = 1 . It is easy to see that (20) has a solution ξ = 1 if and only if λ = ± p (1 + q )(1 + q ) . (The other solution of (20) in this case is ξ = q q .)These are the two eigenvalues of A with the largest magnitude (and each with multiplicity one).Excluding ξ = 1 , which has multiplicity two, means that for λ = − p (1 + q )(1 + q ) , ratherthan four ξ values, we only have two counted in Case A ( ξ = q q with multiplicity two). Thisreduces the total number of eigenvalues ξ for Case A to m + n − N ull ( A )) − . Case B. ( λ i = 0 ) For this case, in the following, we show that we have two eigenvalues ξ i = − q and ξ i = − q for e A . (Note that these eigenvalues are in fact the solutions of (20)for λ i = 0 .) These eigenvalues, based on Lemmas 1 and 5, result in eigenvalues ±√− q and ±√− q for A e . In the following, we also prove that the multiplicities of the eigenvalues ξ i and ξ i of e A are n − Rank ( A ) and m − Rank ( A ) , respectively. This together with Lemma 5prove the claim of the theorem for the multiplicities of eigenvalues ±√− q and ±√− q of A e .To prove that ξ i = − q and ξ i = − q are eigenvalues of e A , and to obtain their multiplicities,we note that the graph G is bipartite, and thus its adjacency matrix has the following form A = n × n D n × m D tm × n m × m . As a result, we have the following fact whose proof is presented in Appendix V-B.
Fact 2.
We have
N ull ( D ) = m − Rank ( A ) / , (26)and N ull ( D t ) = n − Rank ( A ) / . (27)Let −→ µ , . . . , −→ µ t , where t = m − Rank ( A ) / , be the linearly independent eigenvectors ofmatrix D associated with eigenvalue . Corresponding to each vector −→ µ i in the null space of D , we define the following two vectors −→ φ i and −→ φ ′ i , each of size | V | : φ ( x,y ) = a xy µ y , if x ∈ U, y ∈ W,a xy µ x , if x ∈ W, y ∈ U, , otherwise , (28) and φ ′ ( x,y ) = a xy µ y , if x ∈ U, y ∈ W, , otherwise , (29)where φ ( x,y ) ( φ ′ ( x,y ) ) is the element of −→ φ i ( −→ φ ′ i ) corresponding to the pair of nodes ( x, y ) , and µ x ( µ y ) is the element of −→ µ i corresponding to node x ( y ) ∈ W . We then have the followingresult whose proof is provided in Appendix V-C. Fact 3.
Vectors −→ φ i and −→ φ ′ i are eigenvectors of e A associated with eigenvalue − q .Since the vectors −→ µ , . . . , −→ µ t are linearly independent, then by the definitions (28) and (29),the vectors −→ φ , −→ φ ′ , . . . , −→ φ t , −→ φ ′ t are also linearly independent. This implies that the multiplicityof the eigenvalue − q of e A is at least t = 2 m − Rank ( A ) .Similarly, corresponding to each vector −→ µ i , ≤ i ≤ n − Rank ( A ) / , in the null space of D t ,we define the following two vectors −→ φ i and −→ φ ′ i : φ ( x,y ) = a xy µ x , if x ∈ U, y ∈ W,a xy µ y , if x ∈ W, y ∈ U, , otherwise (30)and φ ′ ( x,y ) = a xy µ x , if x ∈ U, y ∈ W, , otherwise . (31)Similar to the proof of Fact 3, it can be seen that these n − Rank ( A ) vectors are eigenvectorsof e A associated with eigenvalue − q . Moreover, they are linearly independent, and thus, themultiplicity of − q is at least n − Rank ( A ) . Finally, the sum of multiplicities of the eigenvalues − q and − q is m + n ) − Rank ( A ) ,which by the Rank-Nullity Theorem, i.e., Rank ( A ) + N ull ( A ) = n + m , is also equal to N ull ( A ) . Case C. ( λ i = 0 and ξ i = 1 ) .In this case, by (20), we have λ i = ± p (1 + q )(1 + q ) . Corresponding to eigenvalue ξ = 1 of e A , we have eigenvalues ± of A e (see, Lemma 5). If the multiplicity of ξ = 1 is m , wehave m/ eigenvalues +1 and m/ eigenvalues − for A e . In Fact 4 that follows, we prove Note that, based on the total multiplicity of the eigenvalues of e A , the multiplicity of the eigenvalues − q and − q of e A is equal to m − Rank ( A ) and n − Rank ( A ) , respectively. that m = 2 | E | − | V | + 2 (proof is given in Appendix V-D). This together with the | V | − eigenvalues η of A e (or ξ of e A ) obtained in Cases A and B, add up to a total number of | E | . Fact 4.
The multiplicity of the eigenvalue ξ = 1 of e A is at least | E | − | V | + 2 . Example 1.
Let G be the complete bipartite graph K m,n . It is well-known that the spectrumof G (eigenvalues of A ) is { m + n − , √ mn, −√ mn } . We thus have N ull ( A ) = m + n − and Rank ( A ) = 2 . We use Theorem 3 to find the eigenvalues of A e .Step 1. The only negative eigenvalue of A is −√ mn . By solving the quadratic equation (20)for λ = −√ mn , we obtain two solutions and ( m − n − . This gives us eigenvalues η = ± p ( m − n − for A e , each with multiplicity one.Step 2. Matrix A e has also eigenvalues ± p − ( m − , each with multiplicity n − Rank ( A ) / n − , and eigenvalues ± p − ( n − , each with multiplicity m − Rank ( A ) / m − .Step 3. Also, A e has eigenvalues ± , each with multiplicity mn − ( m + n ) + 1 .Consequently, using Theorem 1, we have N = 2 mn − m + n ) + 2 + 2 (cid:16) ( m − n − (cid:17) / + (2 n − − m ) / + (2 m − − n ) / × (cid:16) ( m − n − (cid:17) + (cid:16) ( m − n − (cid:17) + ( n − − m ) + ( m − − n )
4= ( m − n − (cid:16) m − n −
1) + ( m −
1) + ( n − (cid:17)
4= ( m − n − mn )4 , (32) and N = 2 mn − m + n ) + 2 + 2 (cid:16) ( m − n − (cid:17) + (2 n − − m ) + (2 m − − n ) m ( m − m − n ( n − n − . (33) Equations (32) and (33) are consistent with the results in the literature [18].
Example 2.
Consider the tesseract graph, denoted by Q , and shown in Fig. 2. This graph, alsoreferred to as the -dimensional hypercube, is bipartite. It is also -regular, and has parameters m = n = 8 , and q = q = 3 . The spectrum of Q is { ( − , ( − , , , } . We use Based on the total number of eigenvalues for e A , the multiplicity of the eigenvalue ξ = 1 is equal to | E | − | V | + 2 . Theorem 3 to find the eigenvalues of A e . From the spectrum of A , we have N ull ( A ) = 6 and Rank ( A ) = 10 .Step 1. Matrix A has two negative eigenvalues: − and − . By solving (20) for λ = − , weobtain two solutions and . This accounts for eigenvalues ± for A e , each with multiplicityone. Also, by solving (20) for λ = − , we obtain two solutions − ± √ i , where i = √− .This accounts for four eigenvalues ± p − ± √ i for A e , each with multiplicity .Step 2. Matrix A e also has eigenvalues ±√− , each with multiplicity n − Rank ( A ) / , andeigenvalues ±√− , each with multiplicity m − Rank ( A ) / ( ±√− , each with multiplicity , in total).Step 3. Also, the matrix A e has the eigenvalues ± , each of multiplicity | E |− ( m + n )+1 = 17 .Now, we use Theorem 1 to find the number of -cycles in Q : N = 2(3) + 8( − √ i ) + 8( − − √ i ) + 12(3) + 348 = 24 . This matches the multiplicity obtained by the backtracking algorithm of [22].
Fig. 2. The tesseract graph Q . IV. C
ONCLUSION
In this paper, we investigated the relationship between the spectra of the adjacency matrix A and the directed edge matrix A e of a bi-regular bipartite graph. We proved that the latter spectrumcan be derived from the former through simple quadratic equations. Through this relationship,we established a connection between two existing computational methods for counting shortcycles (of length less than or equal to g − , where g is the girth of the graph) in bi-regularbipartite graphs. The first method performs such computations using the spectrum of A e and has complexity O ( | E | ) , where | E | is the number of edges in the graph. The second method usesthe graph spectrum and degree sequences of the graph for computations, and has complexity O ( | V | ) , where | V | is the number of nodes in the graph. The latter complexity can be significantlylower than the former for graphs with large node degrees. The downside of the latter approach,however, is that the equations involved in the computations are very tedious, particularly for thecalculation of multiplicity of k -cycles with k ≥ g + 6 . Using the results of this work, one cancompute the multiplicity of short cycles in a bi-regular bipartite graph using the first approachbut with complexity O ( | V | ) (and without any need for the tedious equations of the secondapproach). V. A PPENDIX
A. Proof of Fact 1.
We first prove the following lemma which is subsequently used in the proof of Fact 1.
Lemma 6.
Let G = ( V = U ∪ W, E ) be a bi-regular bipartite graph with adjacency matrix A , andassume that U = { u , . . . , u n } and W = { w , . . . , w m } . If −→ µ ti,j = ( µ i,j,u , . . . , µ i,j,u n , µ i,j,w , . . . , µ i,j,w m ) is an eigenvector of A corresponding to the eigenvalue λ i (index j accounts for the possibilityof multiple eigenvectors corresponding to the same eigenvalue λ i ), and u k ∈ U , then µ i,j,u k = P w t u k ∈ E ( G ) µ i,j,w t λ i . (34)Proof of Lemma 6In the adjacency matrix A of the graph G , sort the nodes in the following order: u , . . . , u n , w , . . . , w m .Let −→ u t = ( µ i,j,u , . . . , µ i,j,u n ) and −→ w t = ( µ i,j,w , . . . , µ i,j,w m ) . Since λ i is an eigenvalue of A wehave: A −→ u −→ w = DD t −→ u −→ w = λ i −→ u −→ w (35)Thus, D −→ w = λ i −→ u and D t −→ u = λ i −→ w . From the first equation, we obtain (34). This completesthe proof of the lemma.To prove Fact 1, we first show that vectors −→ φ i, , . . . , −→ φ i,m i are linearly independent. To provethe claim, we use contradiction. To the contrary, assume that vectors −→ φ i, , . . . , −→ φ i,m i are notlinearly independent. So, there are constant numbers, c i, , . . . , c i,m i , such that at least two arenon-zero and we have c i, −→ φ i, + · · · + c i,m i −→ φ i,m i = −→ . (36) Let xy ∈ E , x ∈ U , and y ∈ W . Consider the row corresponding to the pair of nodes ( x, y ) in(36). We have: c i, φ i, , ( x,y ) + · · · + c i,m i φ i,m i , ( x,y ) = 0 . (37)By substituting (24) in (37) and applying f = λ i ξ i + q , we have m i X j =1 c i,j ( µ i,j,y − λ i ξ i + q µ i,j,x ) = 0 . (38)Since q + 1 ≥ , there is a node y ′ ∈ W , such that y ′ = y and xy ′ ∈ E ( G ) . Similar to (38), wethus have m i X j =1 c i,j ( µ i,j,y ′ − λ i ξ i + q µ i,j,x ) = 0 . (39)From (38) and (21), we obtain m i X j =1 c i,j µ i,j,y = m i X j =1 c i,j µ i,j,y ′ . (40)Since the graph is connected, for any two nodes y, y ′ ∈ W , we have (40). By the same approach,for every two nodes x, x ′ ∈ U , we have λ i ξ i + q m i X j =1 c i,j µ i,j,x = λ i ξ i + q m i X j =1 c i,j µ i,j,x ′ . (41)In Case A, we assumed that λ i = 0 . So, by (20), we have ξ i = − q . Thus, λ i ξ i + q is a nonzeroconstant number. Hence, by (41), we have m i X j =1 c i,j µ i,j,x = m i X j =1 c i,j µ i,j,x ′ . (42)Now, consider the left hand side of (38). By using Lemma 6 for the node x and µ i,j,x , wehave m i X j =1 c i,j (cid:16) µ i,j,y − λ i ξ i + q µ i,j,x (cid:17) = m i X j =1 c i,j (cid:16) µ i,j,y − λ i ξ i + q P y ′ x ∈ E ( G ) µ i,j,y ′ λ i (cid:17) = m i X j =1 c i,j (cid:16) µ i,j,y − P y ′ x ∈ E ( G ) µ i,j,y ′ ξ i + q (cid:17) . (43)By (40), we have m i X j =1 c i,j X y ′ x ∈ E ( G ) µ i,j,y ′ = m i X j =1 c i,j ( q + 1) µ i,j,y . (44) By substituting (44) in (43), we obtain m i X j =1 c i,j (cid:16) µ i,j,y − λ i ξ i + q µ i,j,x (cid:17) = m i X j =1 c i,j (cid:16) µ i,j,y − ( q + 1) µ i,j,y ξ i + q (cid:17) = m i X j =1 c i,j µ i,j,y (cid:16) − q + 1 ξ i + q (cid:17) By (38), we thus have m i X j =1 c i,j µ i,j,y (cid:16) − q + 1 ξ i + q (cid:17) = 0 . (45)Since ξ i = 1 , thus − q +1 ξ i + q = 0 . So, m i X j =1 c i,j µ i,j,y = 0 . (46)By (46) and (38), and since λ i = 0 , we have m i X j =1 c i,j µ i,j,x = 0 . (47)Consequently, m i X j =1 c i,j −→ µ i,j = 0 . (48)This is, however, in contradiction with the eigenvectors −→ µ i, , . . . , −→ µ i,m i being linearly indepen-dent. So, the vectors −→ φ i, , . . . , −→ φ i,m i are linearly independent. With the same approach, we canprove that the vectors −→ ρ i, , . . . , −→ ρ i,m i are linearly independent. B. Proof of Fact 2.
Consider the following adjacency matrix of a bipartite graph G : A = n × n D n × m D tm × n m × m . We have
Rank ( A ) = Rank ( D ) + Rank ( D t ) . (49)Also, Rank ( D ) = Rank ( D t ) . (50)From (49) and (50), we obtain Rank ( A ) = 2 Rank ( D ) . (51) By the Rank-Nullity Theorem for matrix D , we have Rank ( D ) + N ull ( D ) = m . (52)Thus, by (51) and (52), we have N ull ( D ) = m − Rank ( A ) / . Similarly,
N ull ( D t ) = n − Rank ( A ) / . C. Proof of Fact 3.
We show that the vector −→ φ i is an eigenvector of e A associated with eigenvalue − q . Let ξ = 0 be an eigenvalue of e A corresponding to an eigenvector −→ φ . Then, by Lemma 2, φ ( x,y ) = 0 , forany pair of nodes ( x, y ) , where x and y are on the same side of the bipartition. On the otherhand, for ( u, w ) , where u ∈ U and w ∈ W , by (13), we have ξφ ( u,w ) = a uw X x ∈ U a wx X y ∈ W a xy φ ( x,y ) − a uw X x ∈ U a wx φ ( x,w ) − a uw X y ∈ W a uy φ ( u,y ) + a uw φ ( u,w ) . (53)By replacing (28) in the right hand side of (53), we obtain a uw X x ∈ U a wx X y ∈ W a xy µ y − a uw X x ∈ U a wx µ w − a uw X y ∈ W a uy µ y + a uw µ w . (54)Now considering that −→ µ is in the null space of D , the summation P y ∈ W a xy µ y in the firstterm of (54) and P y ∈ W a uy µ y in the third term are zero. The second term of (54) can also besimplified to − a uw ( q + 1) µ w . Thus, Equation (54) reduces to − q a uw µ w , or − q φ ( u,w ) , where φ ( u,w ) is the ( u, w ) th element of −→ φ i , as shown in (28). Similarly, for ( w, u ) , where u ∈ U and w ∈ W , by replacing (28) in the right hand side of (14), and some simplifications, we obtain − q a wu µ w , which is equal to − q φ ( w,u ) , where φ ( w,u ) is the ( w, u ) th element of −→ φ i , as shownin (28). This completes the proof that −→ φ i is an eigenvector of e A associated with the eigenvalue ξ = − q .Similarly, it can be shown that −→ φ ′ i is an eigenvector of e A associated with eigenvalue − q . D. Proof of Fact 4.
To prove the result, we use Lemma 2 to characterize the system of linear equations thatdescribe the eigenvectors of e A associated with the eigenvalue ξ = 1 .First, corresponding to each edge xy ∈ E ( G ) , we define two variables ψ ( x,y ) and ψ ( y,x ) , for atotal of | E | variables. We then define the vector −→ ρ as: ρ ( x,y ) = ψ ( x,y ) , if xy ∈ E ( G ) , , otherwise . (55)Now, for each node u ∈ U , consider the following two linear equations (involving variables ψ ( x,y ) and ψ ( y,x ) ): X y ∈ W a uy ψ ( u,y ) = 0 , (56)and X y ∈ W a uy ψ ( y,u ) = 0 , (57)and for each node w ∈ W , consider the following two linear equations: X x ∈ U a wx ψ ( x,w ) = 0 , (58)and X x ∈ U a wx ψ ( w,x ) = 0 . (59)One can see that if we have the above equations (i.e. (56) and (57) for each u ∈ U , and (58)and (59) for each w ∈ W ), then by (13) and (14), the vector −→ ρ , given in (55), is an eigenvectorof e A associated with eigenvalue ξ = 1 . We note that the total number of equations in (56),(57), (58) and (59) is | V | . From this set of | V | equations, however, at least two are redundant.To show this, consider Equation (58) for a specific node w ∈ W . This equation can be derivedfrom all the remaining equations in (58), and the following equation: X x ∈ U X y ∈ W a xy ψ ( x,y ) = 0 , (60)which itself is obtained by adding up equations in (56) for all the nodes in U . Similarly, oneof the equations in (59) can be deemed redundant, as it can be derived from the rest of theequations in (59), and the equation obtained by adding up all the equations in (57). Having atleast two redundant equations, and removing them from the system of linear equations, we havenow | V | − linear equations and | E | variables. As a result, we have at least | E | − | V | + 2 linearly independent solutions for the eigenvector −→ ρ . R EFERENCES [1] Y. Mao and A. H. Banihashemi, “A heuristic search for good low-density parity-check codes at short block lengths,” in
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