Laplacian Energy of Graphs

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1 Laplacian Energy of Graphs Sigma Xi Research Exhibition Claire T. Carney Library Uniersity of Massachusetts Dartmouth April 28 th & 29 th, 2009 Leanne R. Silia & Gary E. Dais Department of Mathematics A project supported, in part, by the National Science Foundation

2 Matrices associated with a graph Simple graphs consist of points (ertices) joined by lines (edges) with no multiple edges, and no loops. The adjacency matrix of a graph has rows and columns for each ertex i with an entry of 1 in the (i,j) th place if ertices i and j are joined by an edge ( adjacent ) and an entry of 0 otherwise, as shown opposite: We denote by d i the number of edges meeting ertex i (the degree of ertex i ). The normalized Laplacian matrix has an entry when i j, both d i and d i 0, and i is adjacent to j (0 otherwise), and 1 when i = j. So the graph aboe has normalized Laplacian matrix as shown opposite:

3 Eigenalues of the normalized Laplacian matrix Eigenalues of the normalized Laplacian matrix generally reflect deeper properties of a graph, for example: The multiplicity of the eigenalue 0 counts the number of connected components. The largest eigenalue is 2 exactly when the graph is bipartite. The third largest eigenalue roughly measures how hard it is to cut the graph into distinct pieces. The eigenalues of the normalized Laplacian matrix are also related to behaior of random walks on the graph. The eigenalues of the normalized Laplacian of the graph shown before are: 0, 1, 1, 4/3, and 5/3. The eigenalues of the Laplacian matrix always lie between 0 and 2, with the multiplicity of 0 being the number of connected components of the graph (so 0 occurs just once for a connected graph), and with the largest eigenalue = 2 exactly when the graph is bipartite Trees are bipartite: = = A bipartite graph

4 Normalized Laplacian energy We define the normalized Laplacian energy, L(G), of a graph G, as a special instance of the energy of a matrix, namely, if L is the normalized Laplacian matrix of G then where λ 1,, λ n are the eigenalues of L. Note that the 1 in the aboe formula is the aerage of the eigenalues: it is the trace of L diided by the order of L (the number of ertices in the graph). 1 For the graph G = L(G) = / /3-1 = 2

5 Examples of normalized Laplacian energy For the complete graph K n on n ertices, L(K n ) =2, independent of n. This is fairly obious from the form of the normalized Laplacian matrix L: The eigenalues are clearly n/(n-1) with multiplicity n-1 and 0 with multiplicity1, (since these make the rank of λi-l less than n) so L(K n ) = n/(n-1) n/ (n-1)-1 =2 A similar result holds for a star S n with n ertices: L(S n ) =2, independent of n.

6 Normalized Laplacian energy of a path The normalized Laplacian energy L(P n ) of a path P n on n ertices is somewhat harder to compute. The idea to calculate the eigenalues of the normalized Laplacian matrix first relies on calculating the eigenalues of the adjacency matrix, and then carrying out a transformation to get the normalized Laplacian matrix. To calculate the eigenalues of the adjacency matrix of a path, one first does it for a cycle, and then relates a path to a cycle by doubling up the path and joining ends. The result is that the eigenalues of the normalized Laplacian matrix of the path P n on n ertices are: Therefore

7 Maximal and minimal normalized Laplacian energy Based on calculations inoling random graphs (generated using Mathematica s graph theory and combinatorial package Combinatorica) we suspect that: 1. The minimal normalized Laplacian energy of a connected graph is The maximal normalized Laplacian energy for a connected with n ertices is achieed for a path P n on n ertices. In other words, we suspect that for all connected graphs G with n ertices L(G) can increase when an edge is added to a graph: (but not often, we suspect) G 1 = G 2 = L(G 1 ) = L(G 2 ) = 3

8 n # connected graphs with n ertices The number of different connected graphs with n ertices grows rapidly with n, making exhaustie computation of normalized Laplacian energy computationally infeasible een for moderate alues of n Sloane's On-Line Encyclopedia of Integer Sequences sequences/a001349). The 21 different connected graphs with 5 ertices

9 Distribution of normalized Laplacian energy For a gien n, how is the normalized Laplacian energy L(G) of connected graphs G with n ertices distributed? Below are the descriptie statistics of L(G) for connected graphs G with n ertices, for n from 4 through 7: For n = 7, µ and σ are presented as 99% confidence interals; with probability 1, 99% of the time that we sample, a 99% CI will contain the true mean and true standard deiation for all connected graphs with 7 ertices. Histogram of L(G) for 503 random connected graphs G with 7 ertices Aside from the rapidly increasing number of connected graphs, numerically computing eigenalues of the normalized Laplacian matrix of a graph with n ertices is computationally challenging for een moderately large n.