In geometry processing, numerical optimization methods often involve solving sparse linear systems of equations. These linear systems have a structure that strongly resembles to adjacency graphs of the underlying mesh. We observe how classic linear solvers behave on this specific type of problems. For the sake of simplicity, we minimise either the squared gradient or the squared Laplacian, eval...

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...

Let G be a connected graph of order n with Laplacian eigenvalues [Formula: see text]. The Laplacian-energy-like invariant of G, is defined as [Formula: see text]. In this paper, we investigate the asymptotic behavior of the 3.6.24 lattice in terms of Laplacian-energy-like invariant as m, n approach infinity. Additionally, we derive that [Formula: see text], [Formula: see text] and [Formula: see...

Suppose μ1, μ2, ... , μn are Laplacian eigenvalues of a graph G. The Laplacian energy of G is defined as LE(G) = ∑n i=1 |μi − 2m/n|. In this paper, some new bounds for the Laplacian eigenvalues and Laplacian energy of some special types of the subgraphs of Kn are presented. AMS subject classifications: 05C50

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and average degree of the vertices of G. Motivated by the work from Sharafdini et al. [R. Sharafdini, H. Panahbar, Vertex weighted...

Several alternative definitions to graph energy have appeared in literature recently, the first among them being the Laplacian energy, defined by Gutman and Zhou in [Linear Algebra Appl. 414 (2006), 29–37]. We show here that Laplacian energy apparently has small power of discrimination among threshold graphs, by showing that, for each n, there exists a set of n mutually noncospectral connected ...

Abstract The Estrada index of a graph/network is defined as the trace adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many these extensions, including new ones, Gaussian, Mittag–Leffler and Onsager ones. More importantly, contextualize all indices in physico-mathematical frameworks w...

Abstract Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian G,where D(G) are degree diagonal adjacency matrix, respectively. vertex sequence d1 ≥ d2 ≥··· dn let μ1 μ2 μ n−1 &gt;μn = 0 eigenvalues G. invariants, energy (LE), Laplacian-energy-like invariant (LEL) Kirchhoff index (Kf), defined in terms G, as <m:math xmlns:m="http://www.w3.o...

The Laplacian and normalized Laplacian energy of G are given by expressions EL(G) = ∑n i=1 |μi − d|, EL(G) = ∑n i=1 |λi − 1|, respectively, where μi and λi are the eigenvalues of Laplacian matrix L and normalized Laplacian matrix L of G. An interesting problem in spectral graph theory is to find graphs {L,L}−noncospectral with the same E{L,L}(G). In this paper, we present graphs of order n, whi...