Median eigenvalues of bipartite subcubic graphs

It is proved that the median eigenvalues of every connected bipartite graph $G$ of maximum degree at most three belong to the interval $[-1,1]$ with a single exception of the Heawood graph, whose median eigenvalues are $\pm\sqrt{2}$. Moreover, if $G$ is not isomorphic to the Heawood graph, then a positive fraction of its median eigenvalues lie in the interval $[-1,1]$. This surprising result has been motivated by the problem about HOMO-LUMO separation that arises in mathematical chemistry.Comment: Accepted for publication in Combin. Probab. Compu

Median eigenvalues of bipartite graphs

For a graph $G$ of order $n$ and with eigenvalues $\lambda_1\geqslant\cdots\geqslant\lambda_n$, the HL-index $R(G)$ is defined as $R(G) ={\max}\left\{|\lambda_{\lfloor(n+1)/2\rfloor}|, |\lambda_{\lceil(n+1)/2\rceil}|\right\}.$ We show that for every connected bipartite graph $G$ with maximum degree $\Delta\geqslant3$, $R(G)\leqslant\sqrt{\Delta-2}$ unless $G$ is the the incidence graph of a projective plane of order $\Delta-1$. We also present an approach through graph covering to construct infinite families of bipartite graphs with large HL-index

Unsolved Problems in Spectral Graph Theory

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper will be published in Operations Research Transaction

Machine Learning, Quantum Mechanics, and Chemical Compound Space

We review recent studies dealing with the generation of machine learning models of molecular and solid properties. The models are trained and validated using standard quantum chemistry results obtained for organic molecules and materials selected from chemical space at random

Towards the Quantum Machine: Using Scalable Machine Learning Methods to Predict Photovoltaic Efficacy of Organic Molecules

Recent advances in machine learning have resulted in an upsurge of interest in developing a “quantum machine”, a technique of simulating and predicting quantum-chemical properties on the molecular level. This paper explores the development of a large-scale quantum machine in the context of accurately and rapidly classifying molecules to determine photovoltaic efficacy through machine learning. Specifically, this paper proposes several novel representations of molecules that are amenable to learning, in addition to extending and improving existing representations. This paper also proposes and implements extensions to scalable distributed learning algorithms, in order to perform large scale molecular regression. This paper leverages Harvard’s Odyssey supercomputer in order to train various kinds of predictive algorithms over millions of molecules, and assesses cross-validated test performance of these models for predicting photovoltaic efficacy. The study suggests combinations of representations and learning models that may be most desirable in constructing a large-scale system designed to classify molecules by photovoltaic efficacy