Abelian networks III. The critical group

The critical group of an abelian network is a finite abelian group that governs the behavior of the network on large inputs. It generalizes the sandpile group of a graph. We show that the critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network. We exhibit the critical group as a quotient of a free abelian group by a subgroup containing the image of the Laplacian, with equality in the case that the network is rectangular. We generalize Dhar's burning algorithm to abelian networks, and estimate the running time of an abelian network on an arbitrary input up to a constant additive error.Comment: supersedes sections 7 and 8 of arXiv:1309.3445v1. To appear in the Journal of Algebraic Combinatoric

High-ordered spectral characterization of unicyclic graphs

In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let $G$ be a graph and $G^m$ be the $m$-th power (hypergraph) of $G$. The spectrum of $G$ is referring to its adjacency matrix, and the spectrum of $G^m$ is referring to its adjacency tensor. The graph $G$ is called determined by high-ordered spectra (DHS for short) if, whenever $H$ is a graph such that $H^m$ is cospectral with $G^m$ for all $m$, then $H$ is isomorphic to $G$. In this paper we first give formulas for the traces of the power of unicyclic graphs, and then provide some high-ordered cospectral invariants of unicyclic graphs. We prove that a class of unicyclic graphs with cospectral mates is DHS, and give two examples of infinitely many pairs of cospectral unicyclic graphs but with different high-ordered spectra

Conference Program

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications

Random Recursive Hypergraphs

Random recursive hypergraphs grow by adding, at each step, a vertex and an edge formed by joining the new vertex to a randomly chosen existing edge. The model is parameter-free, and several characteristics of emerging hypergraphs admit neat expressions via harmonic numbers, Bernoulli numbers, Eulerian numbers, and Stirling numbers of the first kind. Natural deformations of random recursive hypergraphs give rise to fascinating models of growing random hypergraphs.Comment: 13 pages, 1 figure; v3: minor updates, references adde

Recent developments on the power graph of finite groups - a survey

Funding: Ajay Kumar is supported by CSIR-UGC JRF, New Delhi, India, through Ref No.: 19/06/2016(i)EU-V/Roll No: 417267. Lavanya Selvaganesh is financially supported by SERB, India, through Grant No.: MTR/2018/000254 under the scheme MATRICS. T. Tamizh Chelvam is supported by CSIR Emeritus Scientist Scheme of Council of Scientific and Industrial Research (No.21 (1123)/20/EMR-II), Government of India.Algebraic graph theory is the study of the interplay between algebraic structures (both abstract as well as linear structures) and graph theory. Many concepts of abstract algebra have facilitated through the construction of graphs which are used as tools in computer science. Conversely, graph theory has also helped to characterize certain algebraic properties of abstract algebraic structures. In this survey, we highlight the rich interplay between the two topics viz groups and power graphs from groups. In the last decade, extensive contribution has been made towards the investigation of power graphs. Our main motive is to provide a complete survey on the connectedness of power graphs and proper power graphs, the Laplacian and adjacency spectrum of power graph, isomorphism, and automorphism of power graphs, characterization of power graphs in terms of groups. Apart from the survey of results, this paper also contains some new material such as the contents of Section 2 (which describes the interesting case of the power graph of the Mathieu group M_{11}) and subsection 6.1 (where conditions are discussed for the reduced power graph to be not connected). We conclude this paper by presenting a set of open problems and conjectures on power graphs.Publisher PDFPeer reviewe

Mini-Workshop: Variable Curvature Bounds, Analysis and Topology on Dirichlet Spaces (hybrid meeting)

A Dirichlet form $\mathcal{E}$ is a densely defined bilinear form on a Hilbert space of the form $L^2(X,\mu)$, subject to some additional properties, which make sure that $\mathcal{E}$ can be considered as a natural abstraction of the usual Dirichlet energy $\mathcal{E}(f_1,f_2)=\int_D (\nabla f_1,\nabla f_2)$ on a domain $D$ in $\mathbb{R}^m$. The main strength of this theory, however, is that it allows also to treat nonlocal situations such as energy forms on graphs simultaneously. In typical applications, $X$ is a metrizable space, and the theory of Dirichlet forms makes it possible to define notions such as curvature bounds on $X$ (although $X$ need not be a Riemannian manifold), and also to obtain topological information on $X$ in terms of such geometric information