Maximum spectral radius of outerplanar 3‐uniform hypergraphs

Abstract

In this paper, we study the maximum spectral radius of outerplanar 3‐uniform hypergraphs. Given a hypergraph MJX-tex-caligraphicnormalℋ ${\rm{ {\mathcal H} }}$, the shadow of MJX-tex-caligraphicnormalℋ ${\rm{ {\mathcal H} }}$ is a graph G $G$ with V ( G ) = V ( ℋ ) $V(G)=V({\rm{ {\mathcal H} }})$ and E ( G ) = { u v : u v ∈ h for some h ∈ E ( MJX-tex-caligraphicscriptH ) } $E(G)=\{uv:uv\in \,h\,\text{for\unicode{x02007}some}\,h\in E({\mathscr{H}})\}$. A graph is outerplanar if it can be embedded in the plane such that all its vertices lie on the outer face. A 3‐uniform hypergraph MJX-tex-caligraphicnormalℋ ${\rm{ {\mathcal H} }}$ is called outerplanar if its shadow has an outerplanar embedding such that every hyperedge of MJX-tex-caligraphicnormalℋ ${\rm{ {\mathcal H} }}$ is the vertex set of an interior triangular face of the shadow. Cvetković and Rowlinson conjectured in 1990 that among all outerplanar graphs on n $n$ vertices, the graph K 1 + P n − 1 ${K}_{1}+{P}_{n-1}$ attains the maximum spectral radius. We show a hypergraph analogue of the Cvetković–Rowlinson conjecture. In particular, we show that for sufficiently large n $n$, the n $n$‐vertex outerplanar 3‐uniform hypergraph of maximum spectral radius is the unique 3‐uniform hypergraph whose shadow is K 1 + P n − 1 ${K}_{1}+{P}_{n-1}$.

Spectral Turán-Type Problems on Cancellative Hypergraphs

“…They also obtained a p-spectral version of the Erdős-Ko-Rado theorem on t-intersecting r-graphs. Recently, Ellingham, Lu and Wang [4] showed that the n-vertex outerplanar 3-graph of maximum spectral radius is the unique 3-graph whose shadow graph is the join of an isolated vertex and the path P n−1 . Gao, Chang and Hou [9] studied the spectral extremal problem for K + r+1 -free r-graphs among linear hypergraphs, where K + r+1 is the r-expansion of the complete graph K r+1 , i.e., K + r+1 is obtained from K r+1 by enlarging each edge of K r+1 with (r − 2) new vertices disjoint from V (K r+1 ) such that distinct edges of K r+1 are enlarged by distinct vertices.…”

On the maximum spread of planar and outerplanar graphs

“…There has been extensive research on finding the maximum spectral radius (i.e. the largest eigenvalue) of planar and outerplanar (hyper)graphs and the corresponding extremal (hyper)graphs; see, for example, [3,6,7,9,11,32,17,19,21,28,29,30,31,10]. In [13], Gotshall, O'Brien and Tait studied the maximum spread of outerplanar graphs and narrowed down the structure of the extremal graph attaining the maximum spread.…”