Chemical applications of the Laplacian spectrum. VI On the largest Laplacian eigenvalue of alkanes

The largest Lapacian eigenvalue 1 of the molecular graph is a quantity important in the theory of the photoelectron spectra of saturated hydrocarbons. It is shown that in the case of alkanes, the presence or absence of quaternary carbon atoms is the main structural feature on which 1 depends. Within sets of all alkanes with a given number of carbon atoms the species (with and without quaternary carbons atoms) whose 1-values are minimal and maximal are determined. Let G be a graph with n vertices. The Laplacian matrix of G is defined as L(G )= D(G )- A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees ofG. The eigenvalues ofL(G), denoted by1,2 ,… ,n, form the Laplacian spectrum of G. If they are labeled so that 1 2 …n , then n is always 0, whereas n-1 >0i f and only if the graph G is connected. The largest Laplacian eigenvalue of G is thus 1 = 1 (G). Two recent discoveries in theoretical chemistry resulted in a significant increase of the interest in the Laplacian spectra of molecular graphs. First, the Wiener topological indexW of alkanes was shown to conform to the formula W = n i i n 1