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Forest matrices around the Laplacian matrix
 Linear Algebra and Its Applications
"... We study the matrices Q k of inforests of a weighted digraph Γ and their connections with the Laplacian matrix L of Γ. The (i, j) entry of Q k is the total weight of spanning converging forests (inforests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Q k, can be calc ..."
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Cited by 19 (9 self)
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We study the matrices Q k of inforests of a weighted digraph Γ and their connections with the Laplacian matrix L of Γ. The (i, j) entry of Q k is the total weight of spanning converging forests (inforests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Q k, can
2.2 The Laplacian Matrix
, 2012
"... 2.1 About these notes These notes are not necessarily an accurate representation of what happened in class. The notes written before class say what I think I should say. The notes written after class way what I wish I said. Be skeptical of all statements in these notes that can be made mathematicall ..."
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2.1 About these notes These notes are not necessarily an accurate representation of what happened in class. The notes written before class say what I think I should say. The notes written after class way what I wish I said. Be skeptical of all statements in these notes that can be made mathematically rigorous.
BIPARTITE SUBGRAPHS AND THE SIGNLESS LAPLACIAN MATRIX
"... For a connected graph G, we derive tight inequalities relating the smallest signless Laplacian eigenvalue to the largest normalised Laplacian eigenvalue. We investigate how vectors yielding small values of the Rayleigh quotient for the signless Laplacian matrix can be used to identify bipartite subg ..."
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Cited by 2 (1 self)
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For a connected graph G, we derive tight inequalities relating the smallest signless Laplacian eigenvalue to the largest normalised Laplacian eigenvalue. We investigate how vectors yielding small values of the Rayleigh quotient for the signless Laplacian matrix can be used to identify bipartite
The Third Smallest Eigenvalue Of The Laplacian Matrix
"... Let G be a connected simple graph. The relationship between the third smallest eigenvalue of the Laplacian matrix and the graph structure is explored. For a tree the complete description of the eigenvector corresponding to this eigenvalue is given and some results about the multiplicity of this eige ..."
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Cited by 3 (0 self)
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Let G be a connected simple graph. The relationship between the third smallest eigenvalue of the Laplacian matrix and the graph structure is explored. For a tree the complete description of the eigenvector corresponding to this eigenvalue is given and some results about the multiplicity
Immanantal Polynomials of Laplacian Matrix of Trees
"... The immanant d (\Delta) associated with the irreducible character Ø of the symmetric group S n , indexed by the partition of n, acting on an n \Theta n matrix A = [a ij ] is defined by d (A) = X oe2Sn Ø (oe) n Y i=1 a ioe(i) : For a tree T on n vertices, let L(T ) denote its Laplacian ..."
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Laplacian matrix. Let x be an indeterminate variable and I be the n \Theta n identity matrix. The immanantal polynomial of T corresponding to d is defined as d (xI \Gamma L(T )) = n X k=0 (\Gamma1) k c ;k(T ) x n\Gammak : The coefficients c ;k (T ) admit various algebraic and topological
THE TAU CONSTANT AND THE DISCRETE LAPLACIAN MATRIX OF A Metrized Graph
, 2009
"... We express the tau constant of a metrized graph in terms of the discrete Laplacian matrix and its pseudo inverse. ..."
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Cited by 7 (3 self)
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We express the tau constant of a metrized graph in terms of the discrete Laplacian matrix and its pseudo inverse.
Investigation on spectrum of the adjacency matrix and Laplacian matrix of graph Gl
"... Abstract: Let Gl be the graph obtained from Kl by adhering the root of isomorphic trees T to every vertex of Kl, and dk−j+1 be the degree of vertices in the level j. In this paper we study the spectrum of the adjacency matrix A(Gl) and the Laplacian matrix L(Gl) for all positive integer l, and give ..."
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Abstract: Let Gl be the graph obtained from Kl by adhering the root of isomorphic trees T to every vertex of Kl, and dk−j+1 be the degree of vertices in the level j. In this paper we study the spectrum of the adjacency matrix A(Gl) and the Laplacian matrix L(Gl) for all positive integer l, and give
Results 1  10
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57,210