Spectral extrema of 1-planar graphs (Q6553183)

A graph is \(1\)-planar if it can be drawn in the plane such that each of its edges is crossed at most once. The authors study the spectral radius (i.e., largest eigenvalue of the adjacency matrix) of \(1\)-planar graphs. Firstly, an upper bound \(5+\sqrt{2n+5}\) is given for the spectral radius of an \(n\)-vertex \(1\)-planar graph with \(n\ge 7\). Secondly, the authors prove that a graph uniquely maximizes the spectral radius among all \(n\)-vertex \(1\)-planar graphs for sufficiently large \(n\) (actually for \(n\ge 6.23\times 10^{18}\)). This graph is described as follows: it is the join of \(K_2\) and \(P_{n-2}^{2+}\), where \(P_{n-2}^{2+}\) is the graph obtained from an \((n-2)\)-vertex path \(P_{n-2}\) by adding all possible edges between vertices of distance two and the edge connecting the two end vertices.