where n is the number of vertices of the graph G, and λ1,λ2, . . .,λn are its eigenvalues [1, 4, 5]. Two elementary properties of the graph energy are E(G1 ∪G2) = E(G1) + E(G2) for G1 ∪G2 being the graph consisting of two disconnected components G1 and G2, and E(G∪K1) = E(G), where K1 is the graph with a single vertex. Motivated by the success of the graph-energy concept, and in order to extend...

Gutman et al. introduced the concepts of energy E (G) and Laplacian energy EL(G) for a simple graph G, and furthermore, they proposed a conjecture that for every graph G, E (G) is not more than EL(G). Unfortunately, the conjecture turns out to be incorrect since Liu et al. and Stevanović et al. constructed counterexamples. However, So et al. verified the conjecture for bipartite graphs. In the ...

Let G be a simple graph with order n and size m. The quantity $$M_1(G)=\sum _{i=1}^{n}d^2_{v_i}$$ is called the first Zagreb index of G, where $$d_{v_i}$$ degree vertex $$v_i$$ , for all $$i=1,2,\dots ,n$$ . signless Laplacian matrix $$Q(G)=D(G)+A(G)$$ A(G) D(G) denote, respectively, adjacency diagonal degrees G. $$q_1\ge q_2\ge \dots \ge q_n\ge 0$$ eigenvalues largest eigenvalue $$q_1$$ spectr...

in this paper, we try to find a particular combination of wavelet shrinkage and nonlinear diffusion for noise removal in dental images. we selected the wavelet diffusion and modified its automatic threshold selection by proposing new models for speckle related modulus. the laplacian mixture model and circular symmetric laplacian mixture models were evaluated and as it could be expected, the lat...

This paper presented a simple and efficient algorithm for multi-focus image fusion, which used a multiresolution signal decomposition scheme called Laplacian pyramid method. The principle of Laplacian pyramid transform is introduced, and based on it the fusion strategy is described in detail. The method mainly composed of three steps. Firstly, the Laplacian pyramids of each source image are dec...

We introduce a new operation on a class of graphs with the property that the Laplacian eigenvalues of the input and output graphs are related. Based on this operation, we obtain a family of Θ( √ n) noncospectral unicyclic graphs on n vertices with the same Laplacian energy.

The matrix representations of hypergraphs have been defined via hypermatrices initially. In recent studies, the Laplacian hypergraphs, a generalization matrix, has introduced. this article, based on definition, we derive bounds depending pair-degree, maximum degree, and first Zagreb index for greatest eigenvalue energy r-uniform regular hypergraphs. As result these bounds, Nordhaus–Gaddum type ...

The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It known that solutions to the Dirichlet problem involving such an operator exhibit algebraic boundary singularity regardless domain regularity. This, in turn, deteriorates global regularity and as result convergence rate numerical solutions. For finite element discretizations, we derive local error estimates $H^s$...