Multiplicity of  the second‐largest eigenvalue of a planar graph.

The multiplicity of the second‐largest eigenvalue of the adjacency matrix A(G) of a connected graph G, denoted by m(λ2,G), is the number of times of the second‐largest eigenvalue of A(G) appears. In 2019, Jiang, Tidor, Yao, Zhang, and Zhao gave an upper bound on m(λ2,G) for graphs G with bounded degrees, and applied it to solve a longstanding problem on equiangular lines. In this paper, we show that if G is a 3‐connected planar graph or 2‐connected outerplanar graph, then m(λ2,G)≤δ(G), where δ(G) is the minimum degree of G. We further prove that if G is a connected planar graph, then m(λ2,G)≤Δ(G); if G is a connected outerplanar graph, then m(λ2,G)≤max{2,Δ(G)−1}, where Δ(G) is the maximum degree of G. Moreover, these two upper bounds for connected planar graphs and outerplanar graphs, respectively, are best possible. [ABSTRACT FROM AUTHOR]