The structure of super line graphs. Jay Bagga Daniela Ferrero Department of Computer Science Department of Mathematics Ball State University Texas State University Muncie, IN San Marcos, TX [email protected] [email protected] Abstract A graph G on n vertices {v1,...,vn} can be associated with an adjacency matrix, which is the n × n matrix whose For a given graph G =(V,E) and a positive integer k, entries ai,j are given by ai,j =1if there is an edge joining v v G a =0 the super line graph of index k of G is the graph Sk(G) i and j in and i,j otherwise. For a multigraph, which has for vertices all the k-subsets of E(G), and two ai,j is the number of edges between vi and vj.Thecharac- vertices S and T are adjacent whenever there exist s ∈ S teristic polynomial of the graph G, denoted as χ(G, λ),is and t ∈ T such that s and t share a common vertex. In defined as det(A−λI).Theeigenvalues of the graph G are A α the super line multigraph Lk(G) we have an adjacency for those of .Thealgebraic multiplicity of an eigenvalue each such occurrence. is the multiplicity of α as a root of the characteristic poly- A m (α, A) m (α, G) We give a formula to find the adjacency matrix of Lk(G). nomial of and is denoted as a or a .The If G is a regular graph, we calculate all the eigenvalues of geometric multiplicity of an eigenvalue α is the dimension ker(A − αI) m (α, A) m (α, G) Lk(G) and their multiplicities. From those results we give of and is denoted as g or g . an upper bound on the number of isolated vertices. If A is a real symmetric matrix, every eigenvalue α satisfies ma(α, A)=mg(α, A).Thespectrum of G is the set of eigenvalues of G together with their multiplicities as eigen- values of A. The spectrum of a graph provides valuable 1. Introduction information on its topology. More information on this topic can be found in [8] and [11]. In [4] Bagga, Beineke and Varma introduced the concept We refer the reader to [9] and [10] for background on of super line graphs. For a given graph G =(V,E) and graph concepts not included in this Introduction. a positive integer k,thesuper line graph of index k of G is the graph Sk(G) which has for vertices all the k-subsets 2. Spectral properties of E(G), and two vertices S and T are adjacent whenever s ∈ S t ∈ T s t there exist and such that and share a com- Given two positive integers n and k where n ≥ k,let mon vertex. From the definition, it turns out that S1(G) m = n S m × n k . We denote by n,k the binary matrix coincides with the line graph L(G). Properties of super line whose rows are the m strings with exactly k entries equal graphs were presented in [7], [5] and [2], and a good and to 1.ThenifG is a graph with n edges, the rows of the concise summary can be found in [12]. More specifically, matrix Sn,k represent all the possible k-subsets of edges, or 2 some results regarding the super line graph of index were the vertices of the super line multigraph Lk(G). presented in [6] and [2]. Several variations of the super line graph have been considered. A recent survey of line graphs Lemma 2.1 If G is a graph with n edges, for any integer k, and their generalizations is given in [1]. 1 ≤ k ≤ n, the adjacency matrix of the multigraph Lk(G) In this paper we study the super line multigraph which is A(L (G)) = S A(S )t is defined as follows. For a given graph G =(V,E) and a k n,k n,k k k positive integer ,thesuper line multigraph of index of where A = A(L(G)) is the adjacency matrix of L(G). G is the multigraph Lk(G) whose vertices are the k-subsets of E(G), and two vertices S and T are joined by as many edges as pairs of edges s ∈ S and t ∈ T share a common Corollary 2.2 If G is a graph with n edges, for any inte- vertex. ger k, 1 ≤ k ≤ n, the adjacency matrix of the multigraph Proceedings of the 8th International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN’05) 1087-4089/05 $20.00 © 2005 IEEE t Lk(G) is Obviously, Sn,1 Sn,1 = I. However, if k>1, t Sn,k Sn,k still has a particular pattern. In what follows, for A(L (G)) = S BtB(S )t − 2S (S )t k n,k n,k n,k n,k any matrix X, let us denote by (X)ij the entry in the row i and column j in the matrix X. where B = B(L(G)) is the incidence matrix of G. Lemma 2.4 For an integer n ≥ 2 and an integer k, 1 ≤ Notice that if k>|E| then S (G) has no vertices and if k k<n, let b and c be defined by k = |E|, Sk(G) is a trivial graph with one vertex. For this 1 ≤ k<|E| reason, we focus our attention to the case . n − 2 n − 1 b = and c = Proposition 2.3 Let G be a graph with n edges, k an inte- k − 2 k − 1 1 ≤ k<n A L(G) ger, , and the adjacency matrix of . Let Then m = n k ; then t Sn,k Sn,k = bJ +(c − b)I, χ(S AS t; λ)=(−λ)m−nχ(S tS A; λ). n,k n,k n,k n,k where J is the all-one’s matrix. t Proof: For any eigenvalue α of S AS , define r = t n,k n,k α Proof: S S is a n × n matrix whose entries are m (α, S ASt ) α = n,k n,k a n,k n,k . We first show that for eigenvalues 0 of S ASt , (n) n,k n,k (S tS ) = k (S )t (S ) n,k n,k ij k=1 n,k ik n,k kj r = m (α, S ASt ) (n) α g n,k n,k = k (S ) (S ) k=1 n,k ki n,k kj ≤ m (α, St S A) g n,k n,k ≤ m (α, St S A) Therefore if i = j, (Sn,k)ki and (Sn,k)kj coincide in a n,k n,k n−1 i = j n−2 exactly k−1 positions, and if , in exactly k−2 t S tS The first equality is clear since Sn,kAS is real and positions. Thus the entries of the matrix n,k n,k are all n,k −2 −1 α =0 S ASt b = n , except in the diagonal where all are c = n . symmetric. Suppose is an eigenvalue of n,k n,k. k−2 k−1 v ,...,v Let 1 rα be a basis for the eigenspace corresponding α v v ,...,v to . For any nonzero linear combination of 1 rα , t t t t χ(S tS A; λ) we must have S v =0and S Sn,kAS v = αS v. From Lemma 2.4 we shall obtain n,k n,k , and n,k n,k n,k n,k t m (α, S ASt ) ≤ m (α, St S A) therefore χ(Sn,n−1ASn,n−1 ; λ), for regular graphs. Therefore g n,k n,k g n,k n,k and (λ − α)rα |χ(St S A; λ) n,k n,k . Now consider α =0, let E0 be the eigenspace of S ASt N(St ) n,k n,k corresponding to eigenvalue 0, and let n,k 3. Regular graphs St St n be the null space of n,k. The rank of n,k is , since it is n n e +e easy to see that the linearly independent -vectors i n, In this section we consider a d-regular graph G with n 1 ≤ i ≤ n − 1 e + ···+ e for , and 1 n are in its column edges which has 2(d − 1)-regular line graph L(G).IfA N(St ) m − n space. Therefore the dimension of n,k is . is the adjacency matrix of L(G), then A has eigenvalues t Clearly, N(S ) ⊆ E0. Let v1,...,v ,v − ,...,v n,k mN m n r0 α1 =2(d−1),α2,...,αn with corresponding eigenvectors be a basis for E0 extended from v1,...,v − , a basis m n φ1 = 1,φ2,...,φn. N(St ) v for n,k . Thus any nonzero linear combination v ,...,v St v of m−n+1 r0 yields a nonzero eigenvector n,k Proposition 3.1 Let G be a d-regular graph with n ≥ 2 St S A λr0−m+n|χ(St S A; λ) of n,k n,k , giving that n,k n,k . edges, k an integer, 1 ≤ k<n, and A the adjacency matrix χ(S ASt ; λ)|λm−nχ(St S A; λ) Therefore n,k n,k n,k n,k , and of L(G). Let α1,...,αn be the eigenvalues of A and φi we have equality up to a sign because the polynomials are an eigenvector corresponding to the eigenvalue αi, for i = monic and the degrees are equal.