By Jason J. Molitierno

''Preface at the floor, matrix thought and graph conception are doubtless very diversified branches of arithmetic. despite the fact that, those branches of arithmetic engage because it is usually handy to symbolize a graph as a matrix. Adjacency, Laplacian, and occurrence matrices are well-known to symbolize graphs. In 1973, Fiedler released his first paper on Laplacian matrices of graphs and confirmed what number houses ofRead more...

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**Example text**

3. (See [6]) Let A be a symmetric Z-matrix. Prove that A is a nonsingular M-matrix if and only if A is positive definite. 5 Doubly Stochastic Matrices In this section based largely on [27], we further our study of nonnegative matrices and look at a specific type of nonnegative matrix known as a doubly stochastic matrix. Such matrices are important in the study of Markov chains, probability, and statistics, and will be useful to us when we represent graphs as Laplacian matrices in subsequent chapters.

Observe we have A = P AP ˆ T )n−1 = (P [I + A]P ˆ T )n−1 (I + A)n−1 = (I + P AP ✐ ✐ ✐ ✐ ✐ ✐ “molitierno˙01” — 2011/12/13 — 10:46 — ✐ ✐ 22 Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs n − 1 ˆ2 n − 1 ˆn−1 = P I + (n − 1)Aˆ + A + ... + A PT. 2 n−1 By matrix multiplication, note that Aˆ2 , Aˆ3 , . . , Aˆn−1 all have the same (n − r) × r ˆ Therefore, all of the terms in the square block of 0’s in the lower left corner as A. brackets have an (n−r)×r block of 0’s in the lower left corner, and hence (I +A)n−1 does also.

In this theorem from [41], we present the well-known Courant-Fischer Minimax Theorem. 7 Let A ∈ Mn be symmetric and let k be an integer 1 ≤ k ≤ n. 5) is similar. 5 and fixing k where 2 ≤ k ≤ n, then if x = 0, we have (U T x)T D(U T x) (U T x)T D(U T x) xT Ax = = xT x xT x (U T x)T (U T x). Since U is unitary, we have {U T x : x ∈ Therefore, if w1 , . . ,wn−k n xT Ax xT x n , x = 0} = {y ∈ n : y = 0}. ,U T wn−k ≥ λk . ,wn−k ✐ ✐ ✐ ✐ ✐ ✐ “molitierno˙01” — 2011/12/13 — 10:46 — ✐ Matrix Theory Preliminaries ✐ 13 for any n − k vectors w1 , .