The energy of random graphs

A graph G of order n and size m is called an (n, m)-graph. In what follows we assume that the graph eigenvalues are labeled in a nonincreasing manner, i.e., λ1≥λ2≥⋯≥λ n . If G is connected, then λ1>λ2 [81]. Because λ1≥|λ i |,i=2,…,n, the eigenvalue λ1 is referred to as the spectral radius of G. Three well-known relations for the eigenvalues are 

$$\begin{array}{rcl} & & \sum\limits_{i=1}^{n}{\lambda }_{ i} = 0\end{array}$$

(5.1)



$$\begin{array}{rcl} & & \sum\limits_{i=1}^{n}{\lambda }_{ i}^{2} = 2m\end{array}$$

(5.2)



$$\begin{array}{rcl} & & \sum\limits_{i<j}{\lambda }_{i}\,{\lambda }_{j} = -m.\end{array}$$

(5.3)

The following lemma [81] will be frequently used in the proofs:

/pdf/bounds-for-the-energy-of-graphs-14u9j5lc1q.pdf

Bounds for the Energy of Graphs

Let G be a simple undirected graph. The energy E(G) of G is the sum of the absolute values of the eigenvalues of the adjacent matrix of G, and the Hosoya index Z(G) of G is the total number of matchings in G. A tree is called a nonconjugated tree if it contains no perfect matching. Recently, Ou [‘Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching’, Appl. Math. Lett.19 (2006), 652–656] determined the unique element which is maximal with respect to Z(G) among the family of nonconjugated n-vertex trees in the case of even n. In this paper, we provide a counterexample to Ou’s results. Then we determine the unique maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated n-vertex trees for the case when n is even. As corollaries, we determine the maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated chemical trees on n vertices, when n is even.

/pdf/on-maximal-energy-and-hosoya-index-of-trees-without-perfect-4okbcqpci1.pdf

On maximal energy and hosoya index of trees without perfect matching

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].

Hyperenergetic and Equienergetic Graphs

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].

The Chemical Connection

Let G be a graph on n vertices, and let CHP(G;@l) be the characteristic polynomial of its adjacency matrix A(G). All n roots of CHP(G;@l), denoted by @l"i(i=1,2,...n), are called to be its eigenvalues. The energy E(G) of a graph G, is the sum of absolute values of all eigenvalues, namely, E(G)=@?"i"="1^n|@l"i|. Let U"n be the set of n-vertex unicyclic graphs, the graphs with n vertices and n edges. A fully loaded unicyclic graph is a unicyclic graph taken from U"n with the property that there exists no vertex with degree less than 3 in its unique cycle. Let U"n^1 be the set of fully loaded unicyclic graphs. In this article, the graphs in U"n^1 with minimal and second-minimal energies are uniquely determined, respectively.

Hongbo Hua

Papers

Minimality considerations for graph energy over a class of graphs