The Laplacian polynomial and Kirchhoff index of graphs derived from regular graphs

Let R(G)$R\left(G\right)$ be the graph obtained from G$G$ by adding a new vertex corresponding to each edge of G$G$ and by joining each new vertex to the end vertices of the corresponding edge, and Q(G)$Q\left(G\right)$ be the graph obtained from G$G$ by inserting a new vertex into every edge of G$G$ and by joining by edges those pairs of these new vertices which lie on adjacent edges of G$G$. In this paper, we determine the Laplacian polynomials of R(G)$R\left(G\right)$ and Q(G)$Q\left(G\right)$ of a regular graph G$G$; on the other hand, we derive formulae and lower bounds of the Kirchhoff index of these graphs.     

Analytical solution for a generalized space‐time fractional telegraph equation

In this paper, we consider a nonhomogeneous space‐time fractional telegraph equation defined in a bounded space domain, which is obtained from the standard telegraph equation by replacing the first‐order or second‐order time derivative by the Caputo fractional derivative , α > 0 and the Laplacian operator by the fractional Laplacian ( ? Δ)^{β ∕ 2}, β ∈ (0,2]. We discuss and derive the analytical solutions under nonhomogeneous Dirichlet and Neumann boundary conditions by using the method of separation of variables. The obtained solutions are expressed through multivariate Mittag‐Leffler type functions. Special cases of solutions are also discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Dirac and magnetic Schrödinger operators on fractals

In this paper we define (local) Dirac operators and magnetic Schrödinger Hamiltonians on fractals and prove their (essential) self-adjointness. To do so we use the concept of 1-forms and derivations associated with Dirichlet forms as introduced by Cipriani and Sauvageot, and further studied by the authors jointly with Röckner, Ionescu and Rogers. For simplicity our definitions and results are formulated for the Sierpinski gasket with its standard self-similar energy form. We point out how they may be generalized to other spaces, such as the classical Sierpinski carpet.     

On Spectral Integral Variations of Graphs

Let G be a general graph. The spectrum S ( G ) of G is defined to be the spectrum of its Laplacian matrix. Let G + e be the graph obtained from G by adding an edge or a loop e . We study in this paper when the spectral variation between G and G + e is integral and obtain some equivalent conditions, through which a new Laplacian integral graph can be constructed from a known Laplacian integral graph by adding an edge.     

The property of maximal eigenvectors of trees

Let T be a tree on n vertices and L(T) be its Laplacian matrix. The eigenvalues and eigenvectors of T are respectively referred to those of L(T). With respect to a given eigenvector Y of T, a vertex u of T is called a characteristic vertex if Y [u] = 0 and there is a vertex w adjacent to u with Y [w] ≠ 0; an edge e = (u, w) of G is called a characteristic edge if Y [u]Y [w] < 0. By 𝒞(T, Y) we denote the characteristic set of T with respect to the vector Y, which is defined as the collection of all characteristic vertices and characteristic edges of T corresponding to Y. Merris shows that 𝒞(T, Y) is fixed for all Fiedler vectors of the tree T. An eigenvector of T is called a k-vector (k ≥ 2) of T if this eigenvector corresponds to an eigenvalue λ _k with λ _k > λ _k?1, where λ₁, λ₂, …, λ _n are the eigenvalues of T arranged in non-decreasing order. A k-vector Y of T is called k-maximal if 𝒞(T, Y) has maximum cardinality among all k-vectors of T. We prove that (1) the characteristic set of T with respect to an arbitrary k-vector is contained in that with respect to any k-maximal vector; and consequently (2) the characteristic sets of T with respect to any two k-maximal vectors are same. Our result may be considered as a generalization of Merris' result as Fiedler vectors are 2-maximal.     

Eigenvalues of 2-edge-coverings

A 2-edge-covering between G and H is an onto homomorphism from the vertices of G to the vertices of H so that each edge is covered twice and edges in H can be lifted back to edges in G. In this note we show how to compute the spectrum of G by computing the spectrum of two smaller graphs, namely a (modified) form of the covered graph H and another graph which we term the anti-cover. This is done for both the adjacency matrix and the normalized Laplacian. We also give an example of two anti-cover graphs which have the same normalized Laplacian, and state a generalization for directed graphs.