Abstract
Modelling a chemical compound by a (molecular) graph helps us to obtain some required information about the chemical and physical properties of the corresponding molecular structure. Linear algebraic notions and methods are used to obtain several properties of graphs usually by the help of some graph matrices and these studies form an important sub area of Graph Theory called spectral graph theory. In this paper, we deal with the sum-edge matrices defined by means of vertex degrees. We calculate the sum-edge characteristic polynomials of several important graph classes by means of the corresponding sum-edge matrices.
Highlights
Let G = (V, E) be a graph with | V(G) |= n vertices and | E(G) |= m edges
If u and v are two vertices of G connected by an edge e, this situation is denoted by e = uv
The study of adjacency and incidency with the help of corresponding matrices is a well known application of Graph Theory to Molecular Chemistry and the sub area of Graph Theory dealing with the energy of a graph is called Spectral Graph Theory which uses linear algebraic methods to calculate eigenvalues of a graph resulting in the molecular energy of that graph
Summary
One of these matrices called the sum-edge matrix of G is a square n × n matrix SMe(G) = [aij]n×n determined by the adjacency of vertices as follows: We shall determine this sum-edge characteristic polynomial of some well-known graph classes and give some general results for r-regular graphs.
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