The energy of graphs and matrices

By Vladimir Nikiforov. 2006

349 citations recorded by the source corpus.

Abstract Given a complex m × n matrix A , we index its singular values as σ 1 ( A ) ⩾ σ 2 ( A ) ⩾ ⋯ and call the value E ( A ) = σ 1 ( A ) + σ 2 ( A ) + ⋯ the energy of A , thereby extending the concept of graph energy, introduced by Gutman. Let 2 ⩽ m ⩽ n , A be an m × n nonnegative matrix with maximum entry α , and ‖ A ‖ 1 ⩾ n α . Extending previous results of Koolen and Moulton for graphs, we prove that E ( A ) ⩽ ‖ A ‖ 1 m n + ( m − 1 ) ( ‖ A ‖ 2 2 − ‖ A ‖ 1 2 m n ) ⩽ α n ( m + m ) 2 . Furthermore, if A is any nonconstant matrix, then E ( A ) ⩾ σ 1 ( A ) + ‖ A ‖ 2 2 − σ 1 2 ( A ) σ 2 ( A ) . Finally, we note that Wigner's semicircle law implies that E ( G ) = ( 4 3 π + o ( 1 ) ) n 3 / 2 for almost all graphs G .

Subfields: Computational Theory and Mathematics, Geometry and Topology

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Citations in the corpus

GAN Memory with No Forgetting (2020, methodology): “(b) Influence of truncation (via preserving X% matrix energy [56]) on generation.”
Beyond graph energy: norms of graphs and matrices (2015, background): “10 generalizes Koolen and Moulton’s result (2); inequality (13) is proved in [32], but the characterization of the equality is new.”
Graph energy change due to edge deletion (2008, background): “Hence the energy of a graph is the sum of the singular values of its adjacency matrix [10].”
Spectra of digraphs (2010, background): “Nikiforov [87,88] extended the concept of energy to arbitrary real (or complex) matrices as follows: Let A be an n by m real matrix with, say, n m.”
Randic Matrix and Randic Energy (2010, background): “Nikiforov extended the definition of energy to arbitrary matrices [27], making thus possible to conceive the incidence energy [28,29], based on the incidence matrix, etc.”
Edge addition, singular values, and energy of graphs and matrices (2009, background): “The theorem also shows that what we consider as the energy of a matrix is less than or equal to what already defined as the energy of a matrix in the literature [10]; further, these two energies are coincident for a matrix A if and only if A is normal.”
On incidence energy of a graph (2009, background): “Nikiforov in [20] has extended the concept of graph energy for arbitrary matrices.”
ON LAPLACIAN LIKE ENERGY OF TREES (2011, methodology): “Motivated by Nikiforov’s idea [20], Jooyandeh et al.”
THE NORMALIZED LAPLACIAN MATRIX AND GENERAL RANDI C INDEX OF GRAPHS (2010, background): “Observe that Nikiforov [50] defines the energy of a matrix M of order n to be”
SINGULAR VALUE INEQUALITY AND GRAPH ENERGY CHANGE (2007, background): “Since A(G) is a symmetric matrix, E(G) is indeed the sum of all singular values of A(G), i.e., E(G )= � n i=1 si(A(G)) [ 14 ].”
Graphs and matrices of maximal energy (2006, background): “ions in theoretical chemistry, introduced E (G) = E (A(G)), where A(G) is the adjacency matrix of a graph G. The function E (G) has been studied intensively - see [6] for a survey. Recently Nikiforov [9] showed that if m ≤ n and A ∈ Mm,n is a nonnegative matrix with maximum entry α, then E (A) ≤ (α/”
On the normalized Laplacian energy and general Randić index R-1 of graphs (2010, background): “Along the same lines, the energy of more general matrices and sequences has been studied (see [1, 17]).”
Hermitian-adjacency matrices and Hermitian energies of mixed graphs (2015, neutral): “Consequently, the energy of H ( G ), which is deﬁned as the sum of its singular values [19], is also the sum of the absolute values of its eigenvalues.”