# Smoothness of Graph Energy in Chemical Graphs
## 1 Introduction  
In graph theory and chemical graph theory, graph invariants are quantitative measures that preserve the structural properties of graphs under isomorphism. Graph energy is one of the graph invariants which is associated with the physico-chemical properties of the selected molecules. The origin of this research field goes back 50 years when the famous graph energy was introduced in [1]. Countless papers and some monographs have been written on graph energy; for some of them, see [1; 2; 3]. The monograph [3] is an excellent compilation of results on graph energy, historical background of the subject and methodological approaches, which can serve as a guide for the reader who wishes to know more about this field. Let \(G\) be a (simple) graph with vertex set \(V(G)\) and edge set \(E(G)\). Let the number of vertices of the graph \(G\) be \(n\) and let them be denoted by \(v_{1}\), \(v_{2}\),..., \(v_{n}\). The adjacency matrix \(A(G)\) of the graph \(G\) is a square matrix of order \(n\) whose \((i,j)\) entry is equal to 1 if the vertices \(v_{i}\) and \(v_{j}\) are adjacent and zero otherwise. The eigenvalues \(\lambda_{1}\), \(\lambda_{2}\),..., \(\lambda_{n}\) of the adjacency matrix \(A(G)\) are called the eigenvalues of the graph \(G\)[4]. In [4], [PERSON] et al. state that for molecular graph \(G\) of a conjugated hydrocarbon with eigenvalues \(\lambda_{1}\), \(\lambda_{2}\),..., \(\lambda_{n}\), in the so-called Huckel molecular orbital (HMO) [5] approximation, the energy of the i-th molecular orbital is given by  
\[E_{i}=\alpha+\lambda_{i}\beta,\]  
where \(\alpha\) and \(\beta\) are constants. To simplify the formalism, it is common to set \(\alpha=0\) and \(\beta=1\) so that the \(\pi\)-electron orbital energies and the graph eigenvalues coincide. It follows that the total \(\pi\)-electron energy (\(E\)) is equal to the sum of all \(\pi\)-electron energies present in the molecule, i.e., \(E=\sum_{i=1}^{n}g_{i}E_{i}=\sum_{i=1}^{n}g_{i}\lambda_{i}\), where \(g_{i}\) is the number of electrons in the i-th molecular orbital with energy \(E_{i}\). In [4], it is explained that because of restrictions presented in [6], \(g_{i}\) is 2, 1 or 0. It is important for us to note that in most chemically relevantcases \(g_{i}=2\) when \(\lambda_{i}>0\) and \(g_{i}=0\) when \(\lambda_{i}<0\), which implies that energy is equal to summation over positive eigenvalues \((E=2\sum_{+}\lambda_{i}\), where \(\sum_{+}\) is summation over positive eigenvalues). However, since the sum of all eigenvalues is zero, it follows that  
\[E=E(G)=\sum_{i=1}^{m}|\lambda_{i}|.\]  
The total of \(\pi\)-electron energy and the right-hand side of the previous equation were studied by [PERSON] [7] back in the pioneering days of quantum chemistry. In the 1970s, [PERSON] [1] came up with the idea of defining the energy of a graph \(G\) as the sum of the absolute values of its eigenvalues. In [2], [PERSON] describes a large number of lower and upper bounds for the graph energy, but the first to write a theorem for lower and upper bounds for the graph energy was [PERSON] in 1971 [8]. [PERSON], [PERSON] and [PERSON] introduced an improvement to [PERSON]'s inequality for the total \(\pi\)-electron energy in [9]. [PERSON] and [PERSON] wrote in [10] that the graph energy is a weighted sum of the traces of the even powers of the adjacency matrix. They saw the potential benefit of this finding in new techniques that can be developed to constrain the graph energy where the specific contribution of subgraphs can be determined. This is very important in chemistry, where the search for additional rules for molecular properties contributes to the understanding of such properties in structural terms. [PERSON] and [PERSON] in [11] outline basic principles and facts for the study of graph energies and point out their main applications. A comprehensive research on the subject did not begin until twenty-five years after graph energy was introduced.