Research and Projects
Research and Projects
Regular Dodecahedron : smallest Fullerene Graph.
Imagine a graph as a network of connections. Mathematicians represent graphs as sets: one set, V, contains all the individual points (vertices) in the network, and another set, E, represents the connections (edges) between those points. For instance the skeleton of a regular dodecahedron, a platonic solid, can be visualized as a graph with 20 vertices and 30 edges.
An important tool in analyzing graphs is the adjacency matrix. This matrix captures information about the connections (edges) between individual points (vertices). The eigenvalues of this matrix, collectively called the graph spectrum, hold valuable information about the graph's structure. Spectral Graph Theory asks a fundamental question: can we understand the structure of a network just by studying its spectrum?
Studying the spectral properties of graphs has applications beyond pure mathematics. It has been used in fields like chemistry, theoretical physics, quantum mechanics, and even communication networks. For example, most of the stable naturally occurring fullerene molecules are known to have exactly half of their eigenvalues positive and the other half negative.
One may now wonder, what happens when we restrict the possible values that these eigenvalues can take? Can we still use this spectral information to describe the underlying structure of the graph itself? These are the questions I am trying to answer through my research.
Research Papers
In mathematical chemistry, graphs where each vertex connects to at most three others are often referred to as chemical graphs. In the Hückel molecular orbital model, the median eigenvalues of these graphs correspond to the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO), which are crucial for understanding a molecule's electronic properties. In 2010, Fowler and Pisanski conjectured that, with only a finite number of exceptions, the median eigenvalues of chemical graphs fall within the interval [−1, 1]. They confirmed this for all chemical trees. Building upon this, Mohar in 2013 extended the validation to all bipartite planar chemical graphs and, later in 2016, to all bipartite chemical graphs except the Heawood graph. We fully resolve this conjecture by proving that, except the Heawood graph, the median eigenvalues of all connected chemical graphs lie within [−1, 1]. Additionally, we demonstrate that a positive fraction of the eigenvalues around the median eigenvalues also reside in this interval, mirroring Mohar's findings for bipartite chemical graphs.
A central problem in spectral graph theory is to characterize graphs with bounded adjacency eigenvalues. It is well-known that for the infinite family of line graphs, their adjacency eigenvalues are always greater than or equal to -2. In 1976, Cameron, Goethals, Seidel, and Shult gave a complete characterization of graphs whose smallest eigenvalue is at least -2. Later, work by Hoffman (1972) and Jiang and Polyanskii (2021) showed that x*, the smallest value greater than 2, for which there are infinitely many graphs with the smallest eigenvalue at least -x, is approximately 2.0198. This raises intriguing questions: What makes x* special? What characteristics do graphs with smallest eigenvalues in the range (-x*, -2) exhibit? In our research, we addressed these questions and provided a complete characterization of graphs whose smallest eigenvalue is in the range (-x*,-2). Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in the interval (-x, -2) for any constant x > 2.