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By Jason J. Molitierno

''Preface at the floor, matrix concept and graph idea are probably very assorted branches of arithmetic. besides the fact that, those branches of arithmetic have interaction because it is usually handy to symbolize a graph as a matrix. Adjacency, Laplacian, and prevalence matrices are prevalent to symbolize graphs. In 1973, Fiedler released his first paper on Laplacian matrices of graphs and confirmed what number houses of Read more...

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Writing A as sI − B where B ≥ 0 ˆ where B ˆ is the corresponding submatrix of and s > ρ(B), we see that Aˆ = sI − B ˆ ˆ B. Since s > ρ(B) ≥ ρ(B), it follows that A is also a nonsingular M-matrix. Hence ˆ > 0. by the reasoning of the previous paragraph, det(A) ✷ With some important lemmas in hand, we can now extend our study of Mmatrices to singular M-matrices. We focus on such matrices that are irreducible and observe that the theory of irreducible nonnegative matrices will play an important role in the following theorem from [6].

If the inequality in the above definition is strict, then the M-matrix is nonsingular. The first goal of this section is to investigate the properties of the inverse of a nonsingular M-matrix. To this end, we begin with a lemma from [6] concerning nonnegative matrices. 5 Let T be a nonnegative matrix such that ρ(T ) < 1. 1) converges. 1). Hence (I − T )−1 exists. Observe (I − T )(I + T + T 2 + T 3 + . . + T k ) = I − T k+1 . Since ρ(T ) < 1, it follows that limk→∞ T k+1 = 0. Therefore, taking the limit of both sides as k tends to infinity, we see that ∞ T k = I.

Proof: Let vj = [ 1, . . , 1, 0, . . , 0 ]T be the vector with 1’s as the first j components and zeros elsewhere. Then for j = 1, . . , n, the j-th column of Z is vj . Therefore uij = viT [B(z) − A]vj . 1) Define wj = e − vj . 2) i∈Mj k∈M / j where Mj = {1, . . , j}. Further, since for j = 1, . . 2). However, T Aw vj+1 j T )A(e − v ) = (eT − wj+1 j T Av = n − (n − j − 1) − j + wj+1 j = 1 + vjT Awj+1 , ✐ ✐ ✐ ✐ ✐ ✐ “molitierno˙01” — 2011/12/13 — 10:46 — ✐ ✐ 32 Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs the last equality following from the fact that A is symmetric.

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