1. 1 Some Inequalities on Weighted Vertex Degrees, Eigenvalues, and Laplacian Eigenvalues of Weighted Graphs Behzad Torkian Dept. of Mathematical Sciences University of South Carolina Aiken Advisor : Dr. Rao Li Francis Marion Undergraduate Mathematics Conference Francis Marion University April 11, 2008

2. 2 1. Introduction  A graph G = (V, E).  A weight function W: E -> R, where R is the set of real numbers and W(e) > 0 for each e in E.  The weighted graph G(W) = G(V, E, W).

3. 3  d(v i, G(W)) denotes the weighted degree of vertex v i in G(W). We define d(v i, G(W)) as the sum of weights of edges which are incident with v i. 24

4. 4  We assume that  A(G(W)) is the weighted adjacency matrix of G(W), where if is an edge of G; otherwise.

5. 5  The eigenvalues μ 1 = μ 1 (G(W)) ≥ μ 2 = μ 2 (G(W)) ≥ … ≥ μ n = μ n (G(W)) of A(G(W)) are called the weighted eigenvalues of G(W).  D(G(W)) denotes the diagonal matrix diag[d i ].

6. 6  L(G(W)) = D(G(W)) - A(G(W)) is called weighted Laplacian matrix of G(W).  The eigenvalues λ 1 = λ 1 (G(W)) ≥ λ 2 = λ 2 (G(W)) ≥ … ≥ λ n = λ n (G(W)) of L(G(W)) are called the weighted Laplacian eigenvalues of G(W).

7. 7 2. Result Theorem 1. Let G(W) be a weighted graph of order n such that W(e) > 0 for each edge e of G. Then

8. 8  Theorem 1 generalizes the following Theorem A proved by Rao Li. Theorem A [1]. Let G be a graph of n vertices and e edges. Then

9. 9  Since if W(e) = 1 for each edge e, then c = e and in Theorem 1 become respectively the ordinary vertex degrees and Laplacian eigenvalues in Theorem A.

10. 10 3. A Theorem of Hoffman and Wielandt  If C = A + B, where A, B, and C are symmetric matrices of order n having respectively the eigenvalues arranged in non-increasing order

11. 11 4. Proof Note that the sum of eigenvalues of a matrix M is equal to the sum of diagonal entries of matrix M. Hence

12. 12 For each k,

13. 13 By Cauchy-Schwarz inequality, Note:

14. 14

15. 15 Notice that D(G(W)) = L(G(W)) + A(G(W)). By the theorem of Hoffman and Wielandt, we have that

16. 16 Therefore

17. 17 5. Reference [1]. Rao Li, Some Inequalities on Vertex Degrees, Eigenvalues, and Laplacian Eigenvalues of Graphs, to appear.

18. 18 Thanks.