Relationship Between the Second Largest Adjacency and Signless Laplacian Eigenvalues of Graphs and Properties of Planar Graphs

A graph’s second largest eigenvalue is a significant algebraic characteristic that provides details on the graph’s expansion, connectivity, and randomness. Bounds for the second largest eigenvalue of a graph, denoted as λ2 were previously established in the literature in relation to graph parameters like edge connectivity and vertex connectivity, matching number, independencenumber, and edge expansion constant, among others. A graph is planar if it can be drawn in a plane without graph edges crossing. Determining the planarity of a graph helps in optimizing, simplifying, and understanding complex systems across various fields. Graph skewness, graph thickness, and graph crossing number are a few metrics that describe how much a graph deviates from planarity. In this work, we ascertain the relationship between the graph’s properties, including graph skewness, thickness, and crossing number, and the graph’s second largest eigenvalues of the adjacency matrix A(G) and the signless Laplacian matrix Q(G). Based on the skewness, thickness, and crossing number, we establish a lower bound for the graph’s second largest adjacency and signless Laplacian eigenvalues. We also determine a lower bound for these graph properties in terms of the second largest adjacency and signless Laplacian eigenvalues of regular graphs.