We present a method to derive the complete spectrum of the lift $Γ^α$ of a base digraph $Γ$, with voltage assignments on a (finite) group $G$. The method is based on assigning to $Γ$ a quotient-like matrix whose entries are elements of the group algebra $\mathbb{C}[G]$, which fully represents $Γ^α$. This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of $G$. Thus, our main theorem generalize some previous results of Lováz and Babai concerning the spectra of Cayley digraphs.
In this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show the close relationship between lifted digraphs of voltage digraphs and line digraphs, which are two known ways to obtain dense digraphs. In the same context, we show the equivalence between the vertex-splitting and partial line digraph techniques. Then, we give a sufficient condition for a lifted digraph of a base line digraph to be again a line digraph. Some of the results are illustrated with two well-known families of digraphs. Namely,…
The spectra of digraphs, unlike those of graphs, is a relatively unexplored territory. In a digraph, a separation is a pair of sets of vertices X and Y such that there are no arcs from X and Y . For a subclass of eulerian digraphs, we give an bound on the size of a separation in terms of the eigenvalues of the Laplacian matrix.
Leipzig University（莱比锡大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进研究所数学研究所）；Masaryk University（马萨里克大学）；University of Science and Technology of China（中国科学技术大学）；University of Warwick（华威大学）；University of Bayreuth（拜罗伊特大学）
For a digraph D and three parameters x, y, z in {0,1,+,-} we define the digraph D^(x,y,z) and call it the (x,y,z)-transformation of D. We show that for every r-regular digraph D the adjacency characteristic polynomial A(t, D^(x,y,z)) of (x,y,z)-transformation of D is uniquely defined by r and the adjacency characteristic polynomial A(t, D) of digraph D and we give a description of this function A(t, D^(x,y,z)) = F(r, A(t, D)). We also obtain similar results for some non-regular digraphs, namely, for so-called digraph-functions and their inverse. Also using the (x,y,z)-transformations of digrap…
Thus far, digraphs that are uniquely determined by their Hermitian spectra have proven elusive. Instead, researchers have turned to spectral determination of classes of switching equivalent digraphs, rather than individual digraphs. In the present paper, we consider the traditional notion: a digraph (or mixed graph) is said to be strongly determined by its Hermitian spectrum (abbreviated SHDS) if it is isomorphic to each digraph to which it is cospectral. Convincing numerical evidence to support the claim that this property is extremely rare is provided. Nonetheless, the first infinite family …
The set of distinct eigenvalues of a signed digraph $S$ together with their multiplicities is called its spectrum. The energy of a signed digraph $S$ with eigenvalues $z_1,z_2,\cdots,z_n$ is defined as $E(S)=\sum_{j=1}^{n}|\Re z_j|$, where $\Re z_j $ denotes real part of complex number $z_j$. In this paper, we show that the characteristic polynomial of a bipartite signed digraph of order $n$ with each cycle of length $\equiv 0\pmod 4$ negative and each cycle of length $\equiv 2\pmod 4$ positive is of the form \\ $$ϕ_S(z)=z^n+\sum\limits_{j=1}^{\lfloor{\frac{n}{2}}\rfloor}(-1)^j c_{2j}(S)z^{n-2…
Let $\mathcal{G}_{n,p,q}$ denote the set of strongly connected bipartite digraphs on $n$ vertices which contain a complete bipartite subdigraph $\overleftrightarrow{K_{p,q}}$, where $p, q, n$ are positive integers and $p+q \leq n$. In this paper, we study the Q-index (i.e. the signless Laplacian spectral radius) of $\mathcal{G}_{n,p,q}$, and determine the extremal digraph that has the minimum Q-index.
Distance matrices of graphs were introduced by Graham and Pollack in 1971 to study a problem in communications. Since then, there has been extensive research on the distance matrices of graphs -- a 2014 survey by Aouchiche and Hansen on spectra of distance matrices of graphs lists more than 150 references. In the last ten years, variants such as the distance Laplacian, the distance signless Laplacian, and the normalized distance Laplacian matrix of a graph have been studied. After a brief description of the early history of the distance matrix and its motivating problem, this survey focuses on…
Constructing and studying distributed control systems requires the analysis of the Laplacian spectra and the forest structure of directed graphs. In this paper, we present some basic results of this analysis partially obtained by the present authors. We also discuss the application of these results to decentralized control and touch upon some problems of spectral graph theory.
Associative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.
A random $n$-lift of a base graph $G$ is its cover graph $H$ on the vertices $[n]\times V(G)$, where for each edge $u v$ in $G$ there is an independent uniform bijection $π$, and $H$ has all edges of the form $(i,u),(π(i),v)$. A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan. Let $G$ be a graph with largest eigenvalue $λ_1$ and let $ρ$ be the spectral radius of its universal cover. Friedman (2003) proved that every "new" eigenvalue of a random lift of $G$ is $O(ρ^{1/2}λ_1^{1/2})$ with high probability, …
In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to th…
