Ela a Lower Bound for the Second Largest Laplacian Eigenvalue of Weighted Graphs
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چکیده
, where deg(vi) is the sum of weights of all edges connected to vi. The signless Laplacian matrix Q(G) is defined by D(G) + A(G). We denote by 0 = λ1(G) ≤ λ2(G) ≤ · · · ≤ λn(G) the eigenvalues of L(G), and by μ1(G) ≤ μ2(G) ≤ · · · ≤ μn(G) the eigenvalues of Q(G). We order the degrees of the vertices of G as d1(G) ≤ d2(G) ≤ · · · ≤ dn(G). Various bounds for the Laplacian eigenvalues of unweighted graphs, in terms of their degrees, were studied in the past (e.g., [1]). Li and Pan [6] showed that for an unweighted connected graph G with n ≥ 3, λn−1(G) ≥ dn−1(G). It is interesting to ask whether there exists a similar bound for weighted graphs. We will show it by using the following lemma ([5, p. 178]).
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