Spectral Extremal 1-Planar Graphs with Bounded Pentagon Packing

Abstract

Let $\mathcal P_1$ denote the class of 1-planar graphs and let $tC_5$ be the disjoint union of $t$ copies of $C_5$. For every fixed $t\ge3$ and all sufficiently large $n$, we determine the unique $n$-vertex $tC_5$-free graph in $\mathcal P_1$ with maximum adjacency spectral radius, answering Problem 1 of Li, Wang and Zhao. The proof first gives a structural description of every extremizer. After two dominating vertices are removed, the remainder consists of copies of the seven-vertex graph $B=K_1\vee2K_3$ together with at most one bounded connected core. This follows from a two-family covering theorem for 1-planar $K_2$-joins and the sharp packing--defect inequality \[ 12v(Q)-7e(Q)\ge1-10\nu_5(Q). \] The same inequality yields an exact edge-extremal result for remainders with bounded pentagon packing. A normalized resolvent then cancels the repeated $B$-components, and finite moment comparisons force all packing and all nonzero defect into one core and identify that core uniquely in each residue class modulo $7$. The case $t=3$ is the pentagon-free boundary case and is completed by one exact finite component lemma.