Let G be a graph with V(G) = {1;⋯,n} and E(G) = {e 1,⋯,e m }. The Laplacian matrix of G, denoted by L(G), is the n×n matrix defined as follows. The rows and columns of L(G) are indexed by V(G). If i ≠ j then the (i, j)-entry of L(G) is 0 if vertex i and j are not adjacent, and it is -1 if i and j are adjacent. The (i, i)-entry of L(G) is d i , the degree of the vertex i, i = 1;2,⋯,n.
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References and Further Reading
W.N. Anderson and T.D. Morley, Eigenvalues of the Laplacian of a graph, Linear and Multilinear Algebra, 18(2):141–145 (1985).
R.B. Bapat, Moore-Penrose inverse of the incidence matrix of a tree, Linear and Multilinear Algebra, 42:159–167 (1997).
K. ch. Das, An improved upper bound for Laplacian graph eigenvalues, Linear Algebra Appl., 368:269–278 (2003).
R. Grone and R. Merris, The Laplacian spectrum of a graph, II SIAM J. Discrete Math., 7(2):221–229 (1994).
R. Merris, An edge version of the matrix-tree theorem and the Wiener index, Linear and Multilinear Algebra, 25:291–296 (1989).
J.W. Moon, On the adjoint of a matrix associated with trees, Linear and Multilinear Algebra, 39:191–194 (1995).
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(2010). Laplacian Matrix. In: Graphs and Matrices. Universitext. Springer, London. https://doi.org/10.1007/978-1-84882-981-7_4
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