The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. Gotshall, O'Brien and Tait conjectured that for sufficiently large $n$, the $n$-vertex outerplanar graph with maximum spread is the graph obtained by joining a vertex to a path on $n-1$ vertices. In this paper, we disprove this conjecture by showing that the extremal graph is the graph obtained by joining a vertex to a path on $\lceil(2n-1)/3\rceil$ vertices and $\lfloor(n-2)/3\rfloor$ isolated vertices. For planar graphs, we show that the extremal $n$-vertex planar graph attaining the maximum spread is the graph obtained by joining two nonadjacent vertices to a path on $\lceil(2n-2)/3\rceil$ vertices and $\lfloor(n-4)/3\rfloor$ isolated vertices.
@article{10_37236_11844,
author = {Zelong Li and William Linz and Linyuan Lu and Zhiyu Wang},
title = {On the maximum spread of planar and outerplanar graphs},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {3},
doi = {10.37236/11844},
zbl = {1548.05219},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11844/}
}
TY - JOUR
AU - Zelong Li
AU - William Linz
AU - Linyuan Lu
AU - Zhiyu Wang
TI - On the maximum spread of planar and outerplanar graphs
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/11844/
DO - 10.37236/11844
ID - 10_37236_11844
ER -
%0 Journal Article
%A Zelong Li
%A William Linz
%A Linyuan Lu
%A Zhiyu Wang
%T On the maximum spread of planar and outerplanar graphs
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/11844/
%R 10.37236/11844
%F 10_37236_11844
Zelong Li; William Linz; Linyuan Lu; Zhiyu Wang. On the maximum spread of planar and outerplanar graphs. The electronic journal of combinatorics, Tome 31 (2024) no. 3. doi: 10.37236/11844
