Yongtang Shi

The Chemical Connection

Research on what we call the energy of a graph can be traced back to the 1940s or even to the 1930s. In the 1930s, the German scholar Erich Huckel put forward a method for finding approximate solutions of the Schrodinger equation of a class of organic molecules, the so-called conjugated hydrocarbons. Details of this approach, often referred to as the “Huckel molecular orbital (HMO) theory” can be found in appropriate textbooks [76, 101].

Hyperenergetic and Equienergetic Graphs

The energy of the n-vertex complete graph K n is equal to 2(n − 1). We call an n-vertex graph Ghyperenergetic if ℰ(G) > 2(n − 1). From Theorem 5.24, we know that for almost all graphs, \(\mathcal{E}(G) > \left (\frac{1} {4} + o(1)\right ){n}^{3/2}\), which means that almost all graphs are hyperenergetic. Therefore, any search for hyperenergetic graphs nowadays is a futile task. Yet, before Theorem 5.24 was discovered, a number of such results were obtained. We outline here some of them; for surveys, see [41, 178].

Other Graph Energies

Motivated by the large number of interesting and nontrivial mathematical results that have been obtained for the graph energy, several other energy-like quantities were proposed and studied in the recent mathematical and mathematico–chemical literature. These are listed and briefly outlined in this chapter. Details of the theory of the main alternative graph energies can be found in the references quoted, bearing in mind that research along these lines is currently very active, and new papers/results appear on an almost weekly basis.

The Coulson Integral Formula

In the theory of graph energy, the so-called Coulson integral formula (3.1) plays an outstanding role. This formula was obtained by Charles Coulson as early as 1940 [73] and reads:

Hypoenergetic and Strongly Hypoenergetic Graphs

\mathcal{E}(G) = \frac{1} {\pi }\int\limits_{-\infty }^{+\infty }\left [n -\frac{\mathrm{i}x\,\phi ^{\prime}(G,\mathrm{i}x)} {\phi (G,\mathrm{i}x)} \right ]\mathrm{d}x = \frac{1} {\pi }\int\limits_{-\infty }^{+\infty }\left [n - x \frac{\mathrm{d}} {\mathrm{d}x}\ln \phi (G,\mathrm{i}x)\right ]\mathrm{d}x

Complete solution to a conjecture on the maximal energy of unicyclic graphs

(3.1) where G is a graph, ϕ(G,x) is the characteristic polynomial of G, ϕ′(G,x)=(d∕dx)ϕ(G,x) its first derivative, and \(\mathrm{i} = \sqrt{-1}\).

Note on the energy of regular graphs

After the concept of graph energy was proposed [149], there was much research on this topic. One basic problem is to find the extremal values or the best bounds for the energy within some special classes of graphs and graphs from these classes with extremal values of energy. Finding answers to such questions is often far from elementary. In this chapter we outline some fundamental methods that are frequently used for solving problems of this kind.

Extremal Matching Energy of Bicyclic Graphs

A graph on n vertices, whose energy is less than n, i.e., ℰ(G) < n, is said to be hypoenergetic. Graphs for which ℰ(G) ≥ n are said to be nonhypoenergetic. In [441], a strongly hypoenergetic graph is defined to be a (connected) graph G of order n satisfying ℰ(G) < n − 1. In what follows, for obvious reasons, we assume that all graphs considered are connected.