Spectral graph theory : from practice to theory

Graph theory is the area of mathematics that studies networks, or graphs. It arose from the need to analyse many diverse network-like structures like road networks, molecules, the Internet, social networks and electrical networks. In spectral graph theory, which is a branch of graph theory, matrices are constructed from such graphs and analysed from the point of view of their so-called eigenvalues and eigenvectors. The first practical need for studying graph eigenvalues was in quantum chemistry in the thirties, forties and fifties, specifically to describe the Hückel molecular orbital theory for unsaturated conjugated hydrocarbons. This study led to the field which nowadays is called chemical graph theory. A few years later, during the late fifties and sixties, graph eigenvalues also proved to be important in physics, particularly in the solution of the membrane vibration problem via the discrete approximation of the membrane as a graph. This paper delves into the journey of how the practical needs of quantum chemistry and vibrating membranes compelled the creation of the more abstract spectral graph theory. Important, yet basic, mathematical results stemming from spectral graph theory shall be mentioned in this paper. Later, areas of study that make full use of these mathematical results, thus benefitting greatly from spectral graph theory, shall be described. These fields of study include the P versus NP problem in the field of computational complexity, Internet search, network centrality measures and control theory.peer-reviewe

What is the meaning of the graph energy after all?

For a simple graph $G=(V,E)$ with eigenvalues of the adjacency matrix $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$, the energy of the graph is defined by $E(G)=\sum_{j=1}^{n}|\lambda_{j}|$. Myriads of papers have been published in the mathematical and chemistry literature about properties of this graph invariant due to its connection with the energy of (bipartite) conjugated molecules. However, a structural interpretation of this concept in terms of the contributions of even and odd walks, and consequently on the contribution of subgraphs, is not yet known. Here, we find such interpretation and prove that the (adjacency) energy of any graph (bipartite or not) is a weighted sum of the traces of even powers of the adjacency matrix. We then use such result to find bounds for the energy in terms of subgraphs contributing to it. The new bounds are studied for some specific simple graphs, such as cycles and fullerenes. We observe that including contributions from subgraphs of sizes not bigger than 6 improves some of the best known bounds for the energy, and more importantly gives insights about the contributions of specific subgraphs to the energy of these graphs

An Introductory Note on the Spectrum and Energy of Molecular Graphs

Graph Theory is one branch of Mathematics that laid the foundations of the structural studies in Chemistry. The fact that every molecule or compound can be represented as a network of vertices (elements) and edges (bonds) provoked the question of the predictability of the physical and chemical properties of molecules and compounds. Spectrum, π-electron energy, Spectral Radius etc. are predictable using graph theoretical methods. This is an introductory paper about spectrum and energy of molecular graphs

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

Distributed curvature and stability of fullerenes

Energies of non-planar conjugated π systems are typically described qualitatively in terms of the balance of π stabilisation and the steric strain associated with geometric curvature. Curvature also has a purely graph-theoretical description: combinatorial curvature at a vertex of a polyhedral graph is defined as one minus half the vertex degree plus the sum of reciprocal sizes of the faces meeting at that vertex. Prisms and antiprisms have positive combinatorial vertex curvature at every vertex. Excluding these two infinite families, we call any other polyhedron with everywhere positive combinatorial curvature a PCC polyhedron. Cubic PCC polyhedra are initially common, but must eventually die out with increasing vertex count; the largest example constructed so far has 132 vertices. The fullerenes Cn have cubic polyhedral molecular graphs with n vertices, 12 pentagonal and (n/2 − 10) hexagonal faces. We show that there are exactly 39 PCC fullerenes, all in the range 20 ≤ n ≤ 60. In this range, there is only partial correlation between PCC status and stability as defined by minimum pentagon adjacency. The sum of vertex curvatures is 2 for any polyhedron; for fullerenes the sum of squared vertex curvatures is linearly related to the number of pentagon adjacencies and hence is a direct measure of relative stability of the lower (n ≤ 60) fullerenes. For n ≥ 62, non-PCC fullerenes with a minimum number of pentagon adjacencies minimise mean-square curvature. For n ≥ 70, minimum mean-square curvature implies isolation of pentagons, which is the strongest indicator of stability for a bare fullerene

Estimating the total π-electron energy

The paper gives a short survey of the most important lower and upper bounds for the total π-electron energy, i.e., the graph energy (E). In addition, a new lower and a new upper bound for E are deduced, valid for general molecular graphs. The strengthened versions of these estimates, valid for alternant conjugated hydrocarbons, are also reported. Copyright 2013 (CC) SCS

The Total π-Electron Energy Saga

The total π-electron energy, as calculated within the Hückel tight-binding molecular orbital approximation, is a quantum-theoretical characteristic of conjugated molecules that has been conceived as early as in the 1930s. In 1978, a minor modification of the definition of total π-electron energy was put forward, that made this quantity interesting and attractive to mathematical investigations. The concept of graph energy, introduced in 1978, became an extensively studied graph-theoretical topic, with many hundreds of published papers. A great variety of graph energies is being considered in the current mathematical-chemistry and mathematical literature. Recently, some unexpected applications of these graph energies were discovered, in biology, medicine, and image processing. We provide historic, bibliographic, and statistical data on the research on total π-electron energy and graph energies, and outline its present state of art. The goal of this survey is to provide, for the first time, an as-complete-as-possible list of various existing variants of graph energy, and thus help the readers to avoid getting lost in the jungle of references on this topic. This work is licensed under a Creative Commons Attribution 4.0 International License

The Total π-Electron Energy Saga

The total π-electron energy, as calculated within the Hückel tight-binding molecular orbital approximation, is a quantum-theoretical characteristic of conjugated molecules that has been conceived as early as in the 1930s. In 1978, a minor modification of the definition of total π-electron energy was put forward, that made this quantity interesting and attractive to mathematical investigations. The concept of graph energy, introduced in 1978, became an extensively studied graph-theoretical topic, with many hundreds of published papers. A great variety of graph energies is being considered in the current mathematical-chemistry and mathematical literature. Recently, some unexpected applications of these graph energies were discovered, in biology, medicine, and image processing. We provide historic, bibliographic, and statistical data on the research on total π-electron energy and graph energies, and outline its present state of art. The goal of this survey is to provide, for the first time, an as-complete-as-possible list of various existing variants of graph energy, and thus help the readers to avoid getting lost in the jungle of references on this topic. This work is licensed under a Creative Commons Attribution 4.0 International License