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By Fiedler M., Nedoma J., Ramik J., Rohn J., Zimmermann K.

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If N has a zero row or a zero column then, clearly, det N = 0. This case corresponds to R having an isolated vertex or an edge with both endpoints missing. We assume this not to be the case. Let R be the vertex-disjoint union of the substructures R1 , . . , Rk . After a relabeling of rows and columns if necessary, we have   N1 0 · · · 0  0 N2 0    N= . , ..  ..  . 0 0 Nk where Ni is the incidence matrix of Ri , i = 1, . . , k. 8, we conclude that N is singular. Thus, if Ri has unequal number of vertices and edges for some i then det N = 0.

Are all zero. Therefore, (ii) holds. 12, that (ii) =⇒ (i), and the proof is complete. 3 Bounds We begin with an easy bound for the largest eigenvalue of a graph. 15. Let G be a graph with n vertices, m edges and let λ1 ≥ · · · ≥ λn be 1 )2 . the eigenvalues of G. Then λ1 ≤ ( 2m(n−1) n Proof. As noted earlier, we have ∑ni=1 λi = 0 and ∑ni=1 λi2 = 2m. Therefore, λ1 = − ∑ni=2 λi and hence n λ1 ≤ ∑ |λi |. 1), n 2m − λ12 =∑ i=2 λi2 1 ≥ n−1 n ∑ |λi | 2 ≥ i=2 λ12 . n−1 Hence, 2m ≥ λ12 1 + and therefore λ12 ≤ 1 n−1 = λ12 n n−1 2m(n−1) .

Note that a 0 − 1 vector x of order n × 1 is the incidence vector of a vertex cover if and only if it satisfies M x ≥ 1. 20 and hence is omitted. 21. Let G be a bipartite graph with the incidence matrix M. 2). The following result is the well-known K¨onig–Egervary theorem, which is central to the matching theory of bipartite graphs. 22. If G is a bipartite graph then ν(G) = τ(G). Proof. Let M be the incidence matrix of G. 2) are dual to each other and their feasibility is obvious. Hence, by the duality theorem, their optimal values are equal.

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