## Doctoral Theses

### Perturbed Laplacian Matrix and the Structure of a Graph.

1-28-1999

1-28-2000

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Delhi)

#### Supervisor

Bapat, Ravindra B. (TSMU-Delhi; ISI)

#### Abstract (Summary of the Work)

Laplacian matrices Let G be a connected simple graph with vertex set V = {1,2,.,n), edge set E and let each edge be associated with a positive number, the weight of the edge. The above graph is called a weighted graph. An unweighted graph is just a weighted graph with each of the edges bearing weight 1. All the graphs considered are weighted and simple, unless specified otherwise; all the matrices considered are real. The adjacency matrix A(G) related to this graph is defined as A(G) = (aij), whereaij, if (i, j] â‚¬ E and the weight of the edge is 0, ay = 0, otherwise.Let D be the diagonal matrix with the i-th diagonal entry equal to the sum of the weights of the edges having the vertex i as an end vertex in G. We will call such a matrix as the degree matris of G or simply the degree matrix, when there is no scope of confusion. The Laplacian matriz of G, denoted by L(G), is defined by the = D- A(G). In case there is no scope of confusion we will write L instead of L(G). Let A, S Aa S ... < An be the eigenvalues of L. Let a 1 and suppose that A, A-1. An eigenvector corresponding to the eigenvalue A, of L will be cailed a Piedler s-vector of L(G). The term Fiedler vector will mean a Fiedler 2-vector.In the year 1847, Kirchoff proved the following result involving the Laplacian matrix which put the study of the Laplacian matrix as an interesting subject in front of many researchers. The result is popularly known as the Kirchof's matriz Iree theorem. See [26) to collect some more reforences on this theorem.Theorem Let G be an unweighted graph. Denole by L(ili) the (n-1) x (n- 1) sub- matriz of L obtained by deleting its i-th row and j-th column. Then (-1;+ det L(i|j) is the number of spanning trees in G.In the above theorem det L(ij) means the determinant of the matrix L(ilj). It is understood that if G is disconnected then the number of spanning trees in G is zero. Since then several authors from different disciplines have enriched the subject. The fact that two graphs G and H are isonaorphic if and only if there is a permutation matriz P such that p"LUG)P = L(H), and hence the fact that two graphs are isomorphic only if they have unimodularily congruent Laplacian matrices, do motivate the reader to know more about the Laplacian matrices.Among the studies of different properties and uses of Laplacian matrices the study of Laplacian spectrum and it's relation with the structural properties of graphs has been one of the most attracting features of the subject. To begin with, we can get from the matrix-tree theorem that the rank of L(G) is n- w(G), where w(G) is the number of connected components of G. Thus, assuming that the eigenvalues of L(G) are arranged in nondocrensing order: 0 = A S AS.S An, we see that A = 0 if and only if G is connected. Thus the graph structure is already reflected in the spectrum. This observation led M. Fiedler to define the algebraic connectivily of G by u(G) = 2(G), viewing it as a quantitative measure of connectivity. Fiedler was the first to show that, for a connected graph, further information about the graph structure can be extracted from an eigenvector corresponding to the algebraie con- nectivity by proving some remarkable results. Subsequent observations were made by different authors. We refer the interested readers to (26] for knowing more about the Laplacian matrix and for some references.

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