The Laplacian energy of a graph sums up the absolute values of the differences of average degree and eigenvalues of the Laplace matrix of the graph. This spectral graph parameter is upper bounded by the energy obtained when replacing the eigenvalues with the conjugate degree sequence of the graph, in which the i-th number counts the nodes having degree at least i. Because the sequences of eigen...

This is an expository paper which includes several topics related to the Dirichlet form analysis on the Sierpiński gasket. We discuss the analog of the classical Laplacian; approximation by harmonic functions that gives a notion of a gradient; directional energies and an equipartition of energy; analysis with respect to the energy measure; harmonic coordinates; and non self-similar Dirichlet fo...

The centrality of vertices has been a key issue in network analysis. For unweighted networks where edges are just present or absent and have no weight attached, many centrality measures have been presented, such as degree, betweenness, closeness, eigenvector and subgraph centrality. There has been a growing need to design centrality measures for weighted networks, because weighted networks wher...

Copyright q 2012 X. Pai and S. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let Φ G, λ det λIn − L G ∑n k 0 −1 ck G λn−k be the characteristic polynomial of the Laplacian matrix of a graph G of order n. In this paper, we g...

For a simple connected graph G of order n, having Laplacian eigenvalues μ1, μ2, . . . , μn−1, μn = 0, the Laplacian–energy–like invariant (LEL) and the Kirchhoff index (Kf) are defined as LEL(G) = ∑n−1 i=1 √ μi and Kf(G) = n ∑n−1 i=1 1 μi , respectively. In this paper, LEL and Kf are compared, and sufficient conditions for the inequality Kf(G) < LEL(G) are established.

Let G be a graph of order n with Laplacian spectrum μ1 ≥ μ2 ≥ · · · ≥ μn. The Laplacian-energy-like invariant of graph G, LEL for short, is defined as: LEL(G) = n−1 ∑ k=1 √ μk . In this note, the extremal (maximal and minimal) LEL among all the connected graphs with given matching number is determined. The corresponding extremal graphs are completely characterized with respect to LEL. Moreover ...

Let G be a graph of order n with Laplacian spectrum μ1 ≥ μ2 ≥ · · · ≥ μn. The Laplacian-energy-like invariant of graph G, LEL for short, is defined as: LEL(G) = n−1 ∑ k=1 √ μk . In this note, the extremal (maximal and minimal) LEL among all the connected graphs with given matching number is determined. The corresponding extremal graphs are completely characterized with respect to LEL. Moreover ...

On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy Eφ and conformal Laplacian ∆φ for a given conformal factor φ, based on the corresponding notions in Riemannian geometry in dimension n 6= 2. We derive a differential equation that describes the dependence of the effective resistances of Eφ on φ. We show that the spectrum of∆φ (Dirichlet or Neumann) has si...

We study boundary value problems for the Laplacian on a domain Ω consisting of the left half of the Sierpinski Gasket (SG), whose boundary is essentially a countable set of points X. For harmonic functions we give an explicit Poisson integral formula to recover the function from its boundary values, and characterize those that correspond to functions of finite energy. We give an explicit Dirich...