Wave Equations for Graphs and the Edge-based Laplacian
نویسندگان
چکیده
The main goal of this paper is to develop a “wave equation” for graphs that is very similar to the wave equation utt = ∆u in analysis. Whenever this type of wave equation is involved in a result in analysis, our graph theoretic wave equation seems likely to provide the tool to link the result in analysis to an analogous result in graph theory. Traditional graph theory defines a Laplacian, ∆, as an operator on functions on the vertices. This gives rise to a wave equation utt = −∆u (since graph theory Laplacians are positive semidefinite). However, this wave equation fails to have a “finite speed of wave propagation”. In other words, if u = u(x, t) is a solution, we may have u(x, 0) = 0 for all vertices x within a distance d > 0 to a fixed vertex, x0, without having u(x0, ) vanishing for any > 0. As such, this graph theoretic wave equation cannot link most results in analysis involving the wave equation to a graph theoretic analogue. In this paper we study what appears to be a new type of wave equation on graphs. This wave equation (1) involves a reasonable analogue of utt = ∆u in analysis, (2) has “finite speed of wave propagation” and many other basic properties shared by its analysis counterpart, and (3) seems to be a good vehicle for translating results in analysis to those in graph theory, and vice versa. This wave equation cannot be expressed in the language of traditional graph theory; it requires some of the notions of “calculus on graphs” in [FT99]. It does, however, have a simple physical interpretation—namely, the edges are taut strings, fused together at the vertices. And in fact, the type of Laplacian we use has appeared in the physics literature as the “limiting case” of a “quantum wire” (see [Hur00, RS01, KZ01] for example); but our type of development of the wave equation and its applications to graph theory seem to have escaped the interest of physicists.
منابع مشابه
The Laplacian Polynomial and Kirchhoff Index of the k-th Semi Total Point Graphs
The k-th semi total point graph of a graph G, , is a graph obtained from G by adding k vertices corresponding to each edge and connecting them to the endpoints of edge considered. In this paper, a formula for Laplacian polynomial of in terms of characteristic and Laplacian polynomials of G is computed, where is a connected regular graph.The Kirchhoff index of is also computed.
متن کاملOn net-Laplacian Energy of Signed Graphs
A signed graph is a graph where the edges are assigned either positive ornegative signs. Net degree of a signed graph is the dierence between the number ofpositive and negative edges incident with a vertex. It is said to be net-regular if all itsvertices have the same net-degree. Laplacian energy of a signed graph is defined asε(L(Σ)) =|γ_1-(2m)/n|+...+|γ_n-(2m)/n| where γ_1,...,γ_n are the ei...
متن کاملSTABILITY ANALYSIS FROM FOURTH ORDER NONLINEAR EVOLUTION EQUATIONS FOR TWO CAPILLARY GRAVITY WAVE PACKETS IN THE PRESENCE OF WIND OWING OVER WATER.
Asymptotically exact and nonlocal fourth order nonlinear evolution equations are derived for two coupled fourth order nonlinear evolution equations have been derived in deep water for two capillary-gravity wave packets propagating in the same direction in the presence of wind flowing over water.We have used a general method, based on Zakharov integral equation.On the basis of these evolution eq...
متن کاملSIGNLESS LAPLACIAN SPECTRAL MOMENTS OF GRAPHS AND ORDERING SOME GRAPHS WITH RESPECT TO THEM
Let $G = (V, E)$ be a simple graph. Denote by $D(G)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$ and $A(G)$ the adjacency matrix of $G$. The signless Laplacianmatrix of $G$ is $Q(G) = D(G) + A(G)$ and the $k-$th signless Laplacian spectral moment of graph $G$ is defined as $T_k(G)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...
متن کاملSome results on vertex-edge Wiener polynomials and indices of graphs
The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second ve...
متن کاملThe Signless Laplacian Estrada Index of Unicyclic Graphs
For a simple graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)=sum^{n}_{i=1}e^{q^{}_i}$, where $q^{}_1, q^{}_2, dots, q^{}_n$ are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we first characterize the unicyclic graphs with the first two largest and smallest $SLEE$'s and then determine the unique unicyclic graph with maximum $SLEE$ a...
متن کامل