The Linear Unicyclic Hypergraph with the Second or Third Largest Spectral Radius
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
The Linear Unicyclic Hypergraph with the Second or Third Largest Spectral Radius Chao DING School of Mathematics and Physics, Anqing Normal University, Anqing 246133, P. R. China and School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China E-mail : [email protected]
Yi Zheng FAN1)
Jiang Chao WAN
School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China E-mail : [email protected] [email protected] Abstract The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor of the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph. Keywords
Linear unicyclic hypergraph, adjacency tensor, spectral radius, weighted incident matrix
MR(2010) Subject Classification
1
05C65, 15A18
Introduction
A hypergraph G = (V, E) consists of a nonempty vertex set V = {v1 , v2 , . . ., vn } denoted by V (G) and an edge set E = {e1 , e2 , . . ., em } denoted by E(G), where ei ⊆ V for i ∈ [m] := {1, 2, . . . , m}. If |ei | = k for each i ∈ [m] and k ≥ 2, then G is called a k-uniform hypergraph. In particular, the 2-uniform hypergraphs are exactly the classical simple graphs. The degree of a vertex is the number of edges containing the vertex. A vertex v of G is called a core vertex if it has degree one. An edge e of G is called a pendent edge if it contains |e| − 1 core vertices. Sometimes a core vertex in a pendent edge is also called a pendent vertex. A walk W of length l in G is a sequence of alternate vertices and edges: v0 e1 v1 e2 · · · el vl , where {vi , vi+1 } ⊆ ei for i = 0, 1, . . . , l − 1. If v0 = vl , then W is called a circuit. A walk of G is called a path if no vertices or edges are repeated. A circuit G is called a cycle if no vertices or edges are repeated except v0 = vl . The hypergraph G is said to be connected if every two vertices are connected by a walk. Received February 26, 2020, accepted June 5, 2020 Supported by Natural Science Foundation of China (Grant Nos. 11871073, 11871077) and NSF of Department of Education of Anhui Province (Grant No. KJ2017A362) 1) Corresponding author
Linear Unicyclic Hypergraph
1141
If G is connected and acyclic, then G is called a hypertree (also called supertree in [16] and other literatures). It is known that a k-uniform hypertree on n vertices has n−1 k−1 edges [2, Proposition 4, p. 392]. If G is connected and contains exactly one cycle, then G is called a n unicyclic hypergraph. A k-uniform unicyclic hypergraph on n vertices has k−1 edges [9]. Hu et al. [14] introduced a class of hypergraphs constructed from simple graphs. Let G
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