Graphs with Appended End Vertices: Laplacian Spectra, Laplacian Energy, and Laplacian Eigen-Bi-Balance

In this paper, we determine the Laplacian spectra of graphs obtained by appending  end vertex to all vertices of a defined class of graphs called the base graph. The end vertices allow for a quick solution to the eigen-vector equations of the Laplacian matrix satisfying the characteristic equation, and the solutions to the eigenvalues of the Laplacian matrix of the base graph arise. We determine the relationship between the eigenvalues of the Laplacian matrix of the base graph and the eigenvalues of the Laplacian matrix of the new graph as constructed above, and determine that if  is an eigenvalue of the Laplacian matrix of the base graph, then  is an eigenvalue of the Laplacian matrix of the constructed graph. We then determine the Laplacian spectra for such graphs where the base graph is one of the well-known classes of graphs, namely the complete, complete split-bipartite, cycle, path, wheel and star graphs. We then use the Laplacian spectra to determine the Laplacian energy of the graph, constructed from the base graphs, for each of the above classes of graphs. We then analyse the case where only one end vertex is appended to each vertex in the base graph, and determine the Laplacian energy for large values of  , the total number of vertices in the constructed graph.In the last section, we investigate the eigen-bi-balance of the graphs using the eigenvalues of the Laplacian matrix for graphs with appended end vertices, and consider the example of the star sun graph.