Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method of spectral analysis on self-similar graphs.
We give an axiomatic definition of self-similar graphs which correspond to general nested but not necessarily finitely ramified fractals. For this class of graphs a graph theoretic analogue to the Banach fixed point theorem is proved.
Functional equations and a decomposition algorithm for the Green functions of self-similar graphs with some more symmetric structure are obtained. Their analytic continuations are given by rapidly converging expressions. We study the dynamics of a certain complex rational Green function $d$ on finite directed subgraphs. If the Julia set $\cj$ of $d$ is a Cantor set, then the reciprocal spectrum $\spec^{-1}P=\{1/z\mid z\in\spec P\}$ of the Markov transition operator $P$ can be identified with the set of singularities of any Green function of the whole graph. Finally we get explicit upper and lower bounds for the reciprocal spectrum, where $\cd$ is a countable set of the $d$-backwards iterates of a certain finite set of real numbers.
Discrete Green's functions are the inverses or pseudo-inverses of combinatorial Laplacians. We present compact formulas for discrete Green's functions, in terms of the eigensystems of corresponding Laplacians, for products of regular graphs with or without boundary. Explicit formulas are derived for the cycle, torus, and 3-dimensional torus, as is an inductive formula for the $t$-dimensional torus with $n$ vertices, from which the Green's function can be completely determined in time $O(t n^{2-1/t}\log{n})$. These Green's functions may be used in conjunction with diffusion-like problems on gra…
In this paper, we determine the two normalized Laplacian spectrum of generalized subdivision- vertex corona, subdivision-edge corona for a connected regular graph with an arbitrary reg- ular graph in terms of their normalized Laplacian eigenvalues. Moreover, applying these results, we find some non-regular normaliaed Laplacian cospectral graphs. These results generalize the existing results in [11].
In this paper we compute bounds for signless Laplacian energy, distance signless Laplacian eigenvalues and signless Laplacian energy of unitary addition Cayley graph G_{n}. We also obtain distance Laplacian eigenvalues and distance Laplacian energy of G_{n}.
A centipede is a graph obtained by appending a pendant vertex to each vertex of degree 2 of a path. In this paper we prove that the centipede is determined by its Laplacian spectrum.
We introduce the signless 1-Laplacians and the dual Cheeger constants on simplicial complexes. The connection of its spectrum to the combinatorial properties like independence number, chromatic number and dual Cheeger constant is investigated. Our estimates can be comparable to Hoffman's bounds in virtue of Laplacian on simplicial complexes. An interesting inequality relating multiplicity of the largest eigenvalue, independence number and chromatic number are provided, which could be regarded as a variant version of Lovasz sandwich theorem. Also, the behavior of the operator under the topologi…
The eigenvalues of the normalized Laplacian of a graph provide information on its topological and structural characteristics and also on some relevant dynamical aspects, specifically in relation to random walks. In this paper we determine the spectra of the normalized Laplacian of iterated triangulations of a generic simple connected graph. As an application, we also find closed-forms for their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees.
The study and calculation of spectrum of networks can be used to describe networks structure and quantify analysis of networks performance. The fractal Möbius octagonal networks, denoted by $Q_n$, is derived from the inverse identification of the opposite lateral edges of fractal linear octagonal networks. In this paper, the normalized Laplacian spectrum of $Q_n$ is determined by two matrices $\mathcal {L}_A$ and $\mathcal {L}_S$. As an important application of our results, some topological indices (multiplicative degree-Kirchhoff index, the number of spanning trees) formulas of $Q_n$ are obta…
Let $H_n$ be the linear heptagonal networks with $2n$ heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of $H_n$, we utilize the decomposition theorem. Thus, the Laplacian spectrum of $H_n$ is created by eigenvalues of a pair of matrices: $L_A$ and $L_S$ of order number $5n+1$ and $4n+1$, respectively. On the basis of the roots and coefficients of their characteristic polynomials of $L_A$ and $L_S$, we not only get the explicit forms of Kirchhoff index, but also corresponding total complexity of $H_n$.
The spectral theory of the normalized Laplacian for chemical hypergraphs is further investigated. The signless normalized Laplacian is introduced and it is shown that its spectrum for classical hypergraphs coincides with the spectrum of the normalized Laplacian for bipartite chemical hypergraphs. Furthermore, the spectra of special families of hypergraphs are established.
Let $\mathcal{H}$ be a connected $k$-uniform hypergraph on $n$ vertices and $m$ hyperedges. In [A.~Banerjee, On the spectrum of hypergraph, Linear Algebra and its Application, 614(2021), 82--110], Anirban Banerjee introduced a new adjacency matrix for hypergraphs. In this article we consider the corresponding signless Laplacian matrix $Q(\mathcal{H})$ and discuss about its spectrum.
The power graph $\mathcal{G}(G)$ of a group $G$ is a simple graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if one is a power of other. In this paper, we investigate the Laplacian spectrum of the power graph $\mathcal{G}(\mathbb{Z}_{p^m}^n)$ of finite abelian $p$-group $\mathbb{Z}_{p^m}^n$. In particular, we prove that the spectrum of group $\mathbb{Z}_{p^m}^n$ is contained in the Laplacian spectrum of graph $\mathcal{G}(\mathbb{Z}_{p^m}^n)$. For a finite abelian group $G$ whose power graph $\mathcal{G}(G)$ is planar, we also prove that the spectrum of group…
We show that, in the graph spectrum of the normalized graph Laplacian on trees, the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers. In particular, for the eigenvalue 1, its multiplicity is related to the size of a minimum vertex cover, and zero entries of its eigenvectors correspond to vertices in minimum vertex covers; while for eigenvalues near 1, their distance to 1 can be estimated from minimum vertex covers; and for the largest eigenvalue smaller than 1, the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives.
The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this spectrum under local and global operations like motif doubling, graph joining or splitting. The eigenvalue 1 plays a particular role, and we therefore emphasize those constructions that change its multiplicity in a controlled manner, like the iterated duplication of nodes.
Let Ln denote linear octagonal-quadrilateral networks. In this paper, we aim to firstly investigate the Laplacian spectrum on the basis of Laplacian polynomial of Ln. Then, by applying the relationship between the coefficients and roots of the polynomials, the Kirchhoff index and the complexity are determined.
For some positive integer $k$, if the finite cyclic group $\mathbb{Z}_k$ can act freely on a graph $G$, then we say that $G$ is $k$-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this paper, we investigate a class of 2-connected $k$-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of $k$-symmetric graphs in which all Laplacian eigenvalues are i…
To some extent, graph evolutionary mechanisms can be explained by its spectra. Here, we are interested in two graph operations, namely, motif (subgraph) doubling and attachment that are biologically relevant. We investigate how these two processes affect the spectrum of the normalized graph Laplacian. A high (algebraic) multiplicity of the eigenvalues $1, 1\pm 0.5, 1\pm \sqrt{0.5}$ and others has been observed in the spectrum of many real networks. We attempt to explain the production of distinct eigenvalues by motif doubling and attachment. Results on the eigenvalue $1$ are discussed separate…
Given an arbitrary connected $G$, the $n$-polygon graph $τ_n(G)$ is obtained by adding a path with length $n$ $(n\geq 2)$ to each edge of graph $G$, and the iterated $n$-polygon graphs $τ_n^g(G)$ ($g\geq 0$), is obtained from the iteration $τ_n^g(G)=τ_n(τ_n^{g-1}(G))$, with initial condition $τ_n^0(G)=G$. In this paper, a method for calculating the eigenvalues of normalized Laplacian matrix for graph $τ_n(G)$ is presented if the eigenvalues of normalized Laplacian matrix for graph $G$ is given firstly. Then, the normalized Laplacian spectrums for the graph $τ_n(G)$ and the graphs $τ_n^g(G)$ ($…
Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees…
The present paper is a sequel to our work on hybrid geometry of curves and their moduli spaces. We introduce a notion of hybrid Laplacian, formulate a hybrid Poisson equation, and give a mathematical meaning to the convergence both of the Laplace operator and the solutions to the Poisson equation on Riemann surfaces. As the main theorem of this paper, we then obtain a layered description of the asymptotics of Arakelov Green functions on Riemann surfaces close to the boundary of their moduli spaces. This is done in terms of a suitable notion of hybrid Green functions. As a byproduct of our appr…
