The spectral radius of $1$-planar graphs without complete subgraphs

TL;DR

This paper investigates the spectral properties of 1-planar graphs without certain complete subgraphs, identifying extremal graphs for various clique-free conditions and characterizing their spectral extremal structures.

Contribution

It provides a complete characterization of spectral extremal graphs among 1-planar graphs free of K3, K4, and K5 subgraphs, advancing spectral Turán-type problem understanding.

Findings

Identifies unique spectral extremal graphs for K3- and K4-free 1-planar graphs.

Characterizes spectral extremal graphs for K5-free 1-planar graphs within a small family.

Contributes to spectral Turán-type problems in the context of 1-planar graphs.

Abstract

A 1-planar graph refers to a graph that can be drawn on the plane such that each edge has at most one crossing. In this paper, focusing on the spectral Tur\'{a}n-type problems of $1$-planar graphs, we determine completely the unique spectral extremal graph among all $K_3$-free or $K_4$-free $1$-planar graphs, and provide a characterization of the spectral extremal graphs for $K_5$-free $1$-planar graphs, confining the candidates to a specific, small family.