On minimal forbidden subgraphs for the class of EDM-graphs

Transcription

1 Also available at ISSN printed edn, ISSN 8-97 electronic edn ARS MATHEMATICA CONTEMPORANEA On minimal forbidden subgraphs for the class of EDM-graphs Gašper Jaklič FMF and IMFM, University of Ljubljana, Jadranska 9, 000 Ljubljana, Slovenia and IAM, University of Primorska, Slovenia Jolanda Modic XLAB doo, Pot za Brdom 00, 000 Ljubljana, Slovenia and FMF, University of Ljubljana, Slovenia Received April 0, accepted April 0, published online November 0 Abstract In this paper, a relation between graph distance matrices and Euclidean distance matrices EDM is considered Graphs, for which the distance matrix is not an EDM NEDMgraphs, are studied All simple connected non-isomorphic graphs on n 8 nodes are analysed and a characterization of the smallest NEDM-graphs, ie, the minimal forbidden subgraphs, is given It is proven that bipartite graphs and some subdivisions of the smallest NEDM-graphs are NEDM-graphs, too Keywords: Graph, Euclidean distance matrix, distance, eigenvalue Math Subj Class: A8, 0C0, 0C Introduction A matrix D R n n is Euclidean distance matrix EDM, if there exist x, x,, x n R r, such that d ij = x i x j, i, j =,,, n The minimal possible r is called the embedding dimension see [], eg Corresponding author This research was funded in part by the European Union, European Social Fund, Operational Programme for Human Resources, Development for the Period The author would like to thank XLAB doo, Dr Daniel Vladušič and Dr Gregor Berginc for all the support addresses: gasperjaklic@fmfuni-ljsi Gašper Jaklič, jolandamodic@gmailcom Jolanda Modic cb This work is licensed under

2 Ars Math Contemp Euclidean distance matrices were introduced by Menger in 98 and have received a considerable attention They were studied by Schoenberg [], Young and Householder [], Gower [], and many other authors In recent years many new results were obtained see [, 7, 8, ] and the references therein They are used in various applications in linear algebra, graph theory, geodesy, bioinformatics, chemistry, eg, where frequently a question arises, what can be said about a set of points, if only interpoint distance information is known Some examples can be found in [] EDMs have many interesting properties They are symmetric, hollow ie, with only zeros on the diagonal and nonnegative The sum of their eigenvalues is zero and they have exactly one positive eigenvalue for a nonzero matrix Schoenberg [], Hayden, Reams and Wells [] gave the following characterization of EDMs Theorem Let D R n n be a nonzero symmetric hollow matrix and let e R n be the vector of ones The following propositions are equivalent: a The matrix D is EDM b For all x R n such that x T e = 0, x T Dx 0 c The matrix D has exactly one positive eigenvalue and there exists w R n such that and w T e 0 Dw = e Throughout the paper we will use the notation e for the vector of ones of appropriate size Vectors e i will denote the standard basis Let G be a graph with a vertex set VG and an edge set EG Let the distance du, v between vertices u, v VG be defined as their graph distance, ie, the length of the shortest path between them Let G := [du, v] u,v VG be the distance matrix of G If the graph distance matrix of a graph is EDM, the graph is called an EDM-graph Otherwise the graph is a NEDM-graph Graph distance matrices of EDM-graphs were studied in several papers Path and cycles were analysed in [9] Star graphs and their generalizations were considered in [6, 0] Some results on Cartesian products of EDM-graphs are also known see [] However, the characterization of EDM-graphs in general is still an open problem In this paper, all simple connected non-isomorphic graphs on n 8 nodes are analysed and a characterization of the smallest NEDM-graphs, ie, the minimal forbidden subgraphs, is given In algebraic graph theory, a lot is known on the adjacency matrix and the Laplacian matrix of a graph Many results on their eigenvalues exist, but not much is known on the graph distance matrix Hopefully, this paper will provide a deeper insight into the relation between general graphs or networks and EDM theory There are some interesting possibilities of application Molecular conformation in bioinformatics, dimensionality reduction in statistics, D reconstruction in computer vision, just to name a few The structure of the paper is as follows In Section, all NEDM-graphs on n 8 nodes are considered Analysis of their properties enables us to find some larger NEDM-graphs, which are presented in sections and A proof that bipartite graphs are NEDM-graphs

3 G Jaklič and J Modic: On minimal forbidden subgraphs for the class of EDM-graphs is given We present two families of subdivision graphs of the smallest NEDM-graphs that are NEDM-graphs, too There exist graphs, for which the system has no solution Such graphs are studied in Section The paper is concluded with an example, where we show that not all subdivisions of graphs result in NEDM-graphs The smallest NEDM-graphs In this section we consider simple connected non-isomorphic graphs on n nodes and find the smallest NEDM-graphs There is one simple connected graph on nodes, the path graph P, and there exist only two simple connected graphs on nodes, the path graph P and the cycle graph C In [9] it was proven that path graphs and cycle graphs are EDM-graphs For n =, there are 6 simple connected graphs see Fig First four of them are the star graph S, the path graph P, the cycle graph C and the complete graph K, respectively, which are EDM-graphs see [9, 0] Therefore we only need to consider the last two graphs, G and G 6 6 Figure : Simple connected graphs on nodes Let us denote vertices of graphs G and G 6 counterclockwise by,, and starting with the upper right vertex The characteristic polynomials of the corresponding graph distance matrices 0 0 are G = and G6 = p G λ = λ + λ λ λ 7, λ = λ + λ + λ λ p G 6 Thus matrices G and G 6 have eigenvalues σ G =, 07,, } and σ 6 G = + 7, } 7,, Eigenvalues for G were calculated numerically Exact values can be calculated by using Cardano s formula

4 Ars Math Contemp One can easily verify that vectors w G satisfy the equation G i w G i = [/7, /7, /7, /7] T and w G 6 = [/, 0, /, 0] T = e, i =, 6 Since w T e > 0, i =, 6, by Theorem G i graphs G and G 6 are EDM-graphs Thus there are no NEDM-graphs on nodes In the case n =, there are simple connected graphs see Fig Figure : Simple connected graphs on nodes Graphs G i, i, are the path graph P, the cycle graph C, the complete graph K, the star graph S and the tree T, respectively Since they are EDM-graphs see [], we only need to analyse graphs G i, i = 6, 7,, A straightforward calculation shows that the graph distance matrix G i of the graph G i, i = 6, 7,, 9, has exactly one positive eigenvalue and that there exists w R, G i such that G i w G i = e and w T e 0 By Theorem, graphs G 6 G i, G7,, G9 are EDM-graphs We are left with graphs G 0 and G see Fig The characteristic polynomials of 0 Figure : The graphs G 0 and G

5 G Jaklič and J Modic: On minimal forbidden subgraphs for the class of EDM-graphs the corresponding graph distance matrices 0 G 0 0 = are and G = p 0 G λ = λ + λ 6λ +, λ = λ + λ + λ λ λ + p G Thus matrices G 0 and G have spectra σ G 0 = + 7, 7,,, } and σ G =, 0,,, } Exact eigenvalues for G can be calculated by using Cardano s formula Here they were calculated numerically Since matrices G 0 and G have two positive eigenvalues, graphs G 0 and G are NEDM-graphs These are the smallest NEDM-graphs An induced subgraph H of a graph G is a subset of the vertices VG together with all edges whose endpoints are both in this subset Proposition Let G be a simple connected graph and let H be its induced subgraph If H is a NEDM-graph, the graph G is a NEDM-graph as well Proof Let n and m < n, denote the number of nodes in graphs G and H, respectively Let us order vertices of the graph G in such a way that the first m vertices are the vertices of the graph H Thus the distance matrix G of the graph G is of the form [ ] H G =, where H is the distance matrix of the graph H Every principal submatrix of an EDM has to be an EDM as well Thus since H is not an EDM, neither is G Therefore G is a NEDM-graph All NEDM-graphs form a set of forbidden subgraphs of the class of EDM-graphs Graphs G 0 and G are the minimal forbidden subgraphs All minimal forbidden subgraphs on 6 and 7 nodes can be seen in Fig and Fig Figure : NEDM-graphs for n = 6

6 6 Ars Math Contemp Figure : NEDM-graphs for n = 7 Let mn be the number of NEDM-graphs on n nodes and let m new n be the number of NEDM-graphs on n nodes for which none of the induced subgraphs is NEDM-graph We denote the number of non-isomorphic simple connected graphs on n nodes by gn Table shows how numbers mn and m new n grow with n The calculations were done in the following way By using program geng in Nauty [] we generated all simple connected non-isomorphic graphs on n 8 nodes Then we applied Theorem to determine whether a graph is an EDM-graph Computations were done in Mathematica n gn mn m new n Table : Number of NEDM-graphs compared to the number of all graphs on n nodes Bipartite graphs A quick observation shows that the graph G 0 is bipartite see Fig Let G Uk,Z n k be a simple connected bipartite graph on n nodes, whose vertices are divided into two disjoint sets U k = u, u,, u k }, Z n k = u k+, u k+,, u n }, k =,,, n, such that every edge connects a vertex in U k to a vertex in Z n k see Fig 6 The sets U k and Z n k are called the partition sets A graph join G + G of graphs G and G with disjoint vertex sets VG, VG and edge sets EG, EG is the graph with the vertex set VG VG and the edge set EG EG u, v; u VG, v VG } It is the graph union G G with all the edges that connect the vertices of the first graph with the vertices of the second graph The graph G Uk,Z n k can also be written as the graph join of two empty graphs on k and

7 G Jaklič and J Modic: On minimal forbidden subgraphs for the class of EDM-graphs 7 n k vertices, ie, G Uk,Z n k = O k + O n k The corresponding graph distance matrix is [ ] Ek,k I G k,n k = k E k,n k R n n, E n k,k E n k,n k I n k where E p,q R p q and I p R p p are the matrix of ones and the identity matrix, respectively u k u u u u n u k u k Figure 6: The graph G Uk,Z n k Theorem A simple connected bipartite graph G Uk,Z n k on n nodes and with partition sets U k and Z n k is a NEDM-graph Proof Since graphs G Uk,Z n k and G Un k,z k are isomorphic, it is enough to see that the theorem holds true for k =,,, n/ Let us analyse the eigenvalues of the graph distance matrix of G Uk,Z n k A simple computation shows that u,i = [ e T e T i, ] 0T T solves the equation Gk,n k u,i = u,i for all i =,,, k, and that u,j = [ ] 0 T, e T e T T j, solves the equation Gk,n k u,j = u,j for all j =,,, n k Therefore G k,n k has an eigenvalue with multiplicity n Now let us take u = [ α e T, e ] T T The relation Gk,n k u = λu yields the system of equations which has solutions k α + n k = λ α, kα + n k = λ, α, = k n ± n k + kn k, k λ, = n ± n k + kn k Relations n and k n/ imply that α, and λ, are well-defined Since λ > 0 and λ λ = k n k + n > 0, we conclude that λ > 0 Thus, by Theorem, the graph G Uk,Z n k is a NEDM-graph Remark For k =, the graph G Uk,Z n k is the star graph S n, which is an EDM-graph

8 8 Ars Math Contemp Graph subdivision Let G be a graph A subdivision of an edge in G is a substitution of the edge by a path For example, an edge of the cycle C n can be subdivided into three edges, resulting in the cycle graph C n+ Recall the NEDM-graph G 0 It contains a -cycle c connecting nodes,, and see Fig 7 We can construct larger NEDM-graphs by performing a subdivision of the cycle c Let G 0,n be a graph on n nodes, obtained by subdividing the cycle c in the graph G 0 as seen in Fig 7 Such graphs are G 0,6 = G 6 and G 0,7 = G 7 see Fig and Fig n 0 0, n Figure 7: Construction of graphs G 0,n Let e i denote the standard basis and let C n be the graph distance matrix of the cycle graph C n see [9] The matrix C n is a circulant matrix see [], generated by its first row: 0,,, n 0,,, n We will use the notation C n i,j the matrix C n implies and, n, n,,, n odd,, n, n,,, n even for the i, j-th element of the matrix C n The structure of C, n = C, n =, C, n =, n, C n l, n+/ = n /, l =, n/, l =, n /, l =, Theorem Graphs G 0,n, n, are NEDM-graphs Proof The graph distance matrix of the graph G 0,n, n, is [ ] G 0,n = 0 e T C n + I C n + Ie C n n By Theorem it is enough to show that there exists x R n, such that x T e = 0 and x T G 0,n x > 0

9 G Jaklič and J Modic: On minimal forbidden subgraphs for the class of EDM-graphs 9 Let us take x = [ y T e, y ] T T, with n e + e e + e n+/, n odd, y = n e + n n e e + n n e n+/, n even We will show that x T G 0,n x = yt C n y y T e e T C n y + y T e > 0 From y T e = n, n odd, n +n n, n even, it follows that x T G 0,n x = yt C n y + and y T e = n e T C n y + n n n+ n e T C n y + n n n Firstly, let n be odd Terms in the relation simplify to y T C n y = n C, n C, n + C, n n e T C n y = C,n+/ n n By and, and y T C n y = C,n+/ n n n C, n + C, n x T G 0,n x = n, C,n+/ n n, n odd, n n n, n even,, n odd,, n even, e T C n y = n, + C,n+/ n, which satisfies the requirement for all n When n is even, the terms in the relation simplify to y T n n C n y = n n nc, n nc, n + n C, n + + nc,n+/ n n C,n+/ n e T C n y = n n C,n+/ n n C, By and, n + n n C,n+/ n, C, n y T C n y = n n n n, e T C n y = n

10 60 Ars Math Contemp and x T G 0,n x = n n, which satisfies the requirement for all n Similarly, we can subdivide cycles of the graph G and produce NEDM-graphs see Fig 8 The graph G contains a -cycle c connecting nodes, and Let G,n be a graph on n nodes, obtained by subdividing the cycle c in the graph G as seen in Fig 8 Such graphs are G, = G, G,6 = G 6 and G,7 = G6 7 see Fig and Fig n , n Figure 8: Graphs G,n Theorem Graphs G,n, n, are NEDM-graphs Proof The graph distance matrix of the graph G,n, n, is where G,n = 0 ut 0 v T u v C n u = C n + Ie + e y + n e n+/, v = C n + Ie + y, and y = n+/ k= e k Analogous to the proof of Theorem, we can show that for x = [ α, α, z ] T T, where n, n odd, n α = n, n even, and z =, e n e e n+/, n odd, n e n e e n+/, n even, the expression x T G,n x = zt C n z + α u T z v T z α is positive Relations C, n =, C,n+/ n = C,n+/ n = n, n = n, C,n+/ n = n, C,n+/

11 G Jaklič and J Modic: On minimal forbidden subgraphs for the class of EDM-graphs 6 imply and which yields u T z =, n odd,, n even, z T C n z = Thus by Theorem, the matrix G,n Systems with no solution v T z = n, n odd, n, n even, n n+, n odd, n n, n even, x T G,n x =, n odd,, n even is a NEDM When verifying whether a graph G with the corresponding graph distance matrix G is an EDM-graph, by Theorem one can check if there exists a solution of the equation Gw = e, such that w T e 0 For n 7 there exist graphs, for which the equation Gw = e has no solution Let G k,n k be the graph join of a complete graph K k and an empty graph O n k, n 7, k =,,, n, ie, G k,n k = K k + O n k The graph K k contains vertices,,, k and the graph O n k contains vertices k +, k +,, n Thus the corresponding graph distance matrix is [ ] Ek,k I G k,n k = k E k,n k E n k,k E n k,n k I n k For n = 7 and k = the equation G, w = e has no solution since the ranks of the matrix G, and its augmented matrix [G, e] are different, rankg, = 6 and rank[g, e] = 7 The same holds true if n = 7 and k = Thus by Theorem matrices G, and G, are not EDMs On the other hand, for n = 8 the equation G k,8 k w = e has a solution for all k,, } In general, the matrix G k,n k is a NEDM Theorem The graph K k + O n k, n 7, k =,,, n, is a NEDM-graph Proof Let G k,n k be the graph distance matrix of the graph K k + O n k, n 7, k =,,, n For k = we take w = [ n, n,,,, ] T We can verify that G,n w = e and w T e = 6 n/ < 0 Thus by Theorem the matrix G,n is a NEDM Now let k =,,, n For n = 7 the proof has already been done above For n 8 let u = [ α e T, e T ] T, where vectors e are of sizes k and n k, respectively The relation G k,n k u = λu yields the system of equations αk + n k = λα, αk + n k = λ,

12 6 Ars Math Contemp with solutions α, = k n + ± n kn k + k + k, λ, = n k ± n kn k + k + Relations n 8 and k n imply that α, and λ, are well-defined Since λ > 0 and λ λ = n kk + n 7 > 0, we conclude that λ > 0 Thus, by Theorem, graph K k +O n k is a NEDM-graph Remark For k = and k = n, the graphs K k + O n k are the star graph S n and the complete graph K n, respectively, which are EDM-graphs Remark For k = n, the graph K n + O is an EDM-graph The graph distance matrix G n, has eigenpairs, [ 0 T, e T e T ] T,, [ e T e T i, 0 T ] T, i =,,, n, and λ,, [ α, e T, e T ] T with α, = n ± n n + 9 n and λ, = n ± n n + 9 The eigenvalue λ is obviously positive From λ λ = it follows that λ < 0 One can easily verify that w = / [ 0 T, e T ] T solves the equation Gn, w = e Since w T e =, Theorem implies that G n, is EDM 6 Conclusion In Section we studied subdivisions of graphs Not all graph subdivisions result in NEDMgraphs Consider subdividing graph G 0 as in Fig 9 and denoting it by H The corresponding graph distance matrix H = has eigenvalues σ H = 0, 0, 0, 06,,, }, which were calculated numerically Exact eigenvalues could be obtained using Cardano s formula One can easily verify that vector w H = [/, /, /, /, /, 0, 0] T solves the equation Hw H = e Since w T He = /, by Theorem the graph H is an EDM-graph

13 G Jaklič and J Modic: On minimal forbidden subgraphs for the class of EDM-graphs Figure 9: A subdivision of the graph G 0 References [] R Balaji and R B Bapat, Block distance matrices, Electron J Linear Algebra 6 007, [] J Dattorro, Convex Optimization and Euclidean Distance Geometry, Meboo, 008 [] G H Golub and CF van Loan, Matrix computations, Johns Hopkins Studies in the Mathematical Sciences, third ed, Johns Hopkins University Press, 996 [] J C Gower, Euclidean distance geometry, Math Sci 7 98, [] T L Hayden, R Reams and J Wells, Methods for constructing distance matrices and the inverse eigenvalue problem, Linear Algebra Appl 9 999, 97 [6] G Jaklič, J Modic, On properties of cell matrices, Appl Math Comput 6 00, 06 0 [7] G Jaklič, J Modic, A note on Methods for constructing distance matrices and the inverse eigenvalue problem, Linear Algebra Appl 7 0, [8] G Jaklič and J Modic, Inverse eigenvalue problem for Euclidean distance matrices of size, Bull Aust Math Soc 87 0, 8 9 [9] G Jaklič and J Modic, On Euclidean distance matrices of graphs, Electron J Linear Algebra 6 0, 7 89 [0] G Jaklič, J Modic, Euclidean graph distance matrices of generalizations of the star graph, Appl Math Comput 0 0, [] G Jaklič, J Modic, Cartesian products of EDM-graphs, submitted [] B D McKay, Practical Graph Isomorphism, Congr Numer 0 98, 87 [] I J Schoenberg, Metric spaces and positive definite functions, Trans Amer Math Soc 98, 6 [] G Young, A Householder, Discussion of a set of points in terms of their mutual distances, Psychometrika 98, 9