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The Laplacian spectrum of graphs
 Graph Theory, Combinatorics, and Applications
, 1991
"... Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, m ..."
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Cited by 148 (1 self)
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Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidthtype parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,
Binomial Coefficients and Characters of the Symmetric Group
, 1996
"... this article we suggest a family of weights a k , derived from the irreducible characters of the symmetric group, that would include (1) and (2) as special cases. This generalization provides much insight into the nature of the identities (1) and (2), and enables us to appreciate some of the proper ..."
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Cited by 2 (2 self)
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this article we suggest a family of weights a k , derived from the irreducible characters of the symmetric group, that would include (1) and (2) as special cases. This generalization provides much insight into the nature of the identities (1) and (2), and enables us to appreciate some of the properties such as why the Email: matlamtk@math.nus.sg L(G) = B B B B B B B B B B B \Gamma1 1 0 \Gamma1 1 . .
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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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 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 interpretations for the tree T . We study the properties of c ;k (T ) as well as upper and lower bounds on c ;k (T ) and in particular show that c ;k (S(n)) c ;k (T ) c ;k (P (n)) for all partitions and 0 k n, where S(n) and P (n) denote the star and the path on n vertices respectively. This answers some questions posed in [16] in the affirmative. We study the properties of c ;k (T ) in more detail w...
Vertex Orientations and Immanants of Bipartite Graphs
, 1997
"... In [2], Brualdi and Goldwasser obtained lower bounds for the permanent of the Laplacian of bipartite graphs and trees with given diameter, trees with given bipartition, and trees with given matching size. Some of these results were extended in [7] to other immanants of the Laplacian of trees using a ..."
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In [2], Brualdi and Goldwasser obtained lower bounds for the permanent of the Laplacian of bipartite graphs and trees with given diameter, trees with given bipartition, and trees with given matching size. Some of these results were extended in [7] to other immanants of the Laplacian of trees using a new interpretation for the matching numbers. In this work, we study immanantal inequalities for bipartite graphs using the concept of vertex orientation discussed in [7] and inequalitites involving sums of irreducible characters. 1 Introduction Let S n be the permutation group on n symbols. Let Ø be an irreducible character of S n indexed by a partition . The immanant function, d associated with the character Ø acting on an n \Theta n matrix A = [a ij ], is defined as d (A) = X oe2Sn Ø (oe) n Y i=1 a ioe(i) : If = (n), then Ø (n) is the trivial character and d (n) is just the permanent function, denoted as per : For = (1 n ), Ø (1 n ) is the alternating character and d (...
EDGEGRAFTING THEOREMS ON PERMANENTS OF THE LAPLACIAN MATRICES OF GRAPHS AND THEIR APPLICATIONS
, 2013
"... The trees, respectively unicyclic graphs, on n vertices with the smallest Laplacian permanent are studied. In this paper, by edgegrafting transformations, the nvertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the nvertex bipart ..."
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The trees, respectively unicyclic graphs, on n vertices with the smallest Laplacian permanent are studied. In this paper, by edgegrafting transformations, the nvertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the nvertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian permanent are characterized. Consequently, the nvertex bipartite unicyclic graphs with the first, second and third smallest Laplacian permanent are determined.