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Article overview
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The largest $H$-eigenvalue and spectral radius of Laplacian tensor of non-odd-bipartite generalized power hypergraphs | | Yi-Zheng Fan
; Murad-ul-Islam Khan
; Ying-Ying Tan
; | | Date: |
8 Oct 2015 | | Abstract: | Let $G$ be a simple graph or hypergraph, and let $A(G),L(G),Q(G)$ be the
adjacency, Laplacian and signless Laplacian tensors of $G$ respectively. The
largest $H$-eigenvalues (resp., the spectral radii) of $L(G),Q(G)$ are denoted
respectively by $lambda_{max}^L(G), lambda_{max}^Q(G)$ (resp., $
ho^L(G),
ho^Q(G)$). For a connected non-bipartite simple graph $G$,
$lambda_{max}^L(G)=
ho^L(G) <
ho^Q(G)$. But this does not hold for
non-odd-bipartite hypergraphs. We will investigate this problem by considering
a class of generalized power hypergraphs $G^{k,frac{k}{2}}$, which are
constructed from simple connected graphs $G$ by blowing up each vertex of $G$
into a $frac{k}{2}$-set and preserving the adjacency of vertices.
Suppose that $G$ is non-bipartite, or equivalently $G^{k,frac{k}{2}}$ is
non-odd-bipartite. We get the following spectral properties: (1)
$
ho^L(G^{k,{k over 2}}) =
ho^Q(G^{k,{k over 2}})$ if and only if $k$ is a
multiple of $4$; in this case
$lambda_{max}^L(G^{k,frac{k}{2}})<
ho^L(G^{k,{k over 2}})$. (2) If
$kequiv 2 (!!!mod 4)$, then for sufficiently large $k$,
$lambda_{max}^L(G^{k,frac{k}{2}})<
ho^L(G^{k,{k over 2}})$. Motivated by
the study of hypergraphs $G^{k,frac{k}{2}}$, for a connected non-odd-bipartite
hypergraph $G$, we give a characterization of $L(G)$ and $Q(G)$ having the same
spectra or the spectrum of $A(G)$ being symmetric with respect to the origin,
that is, $L(G)$ and $Q(G)$, or $A(G)$ and $-A(G)$ are similar via a complex
(necessarily non-real) diagonal matrix with modular-$1$ diagonal entries. So we
give an answer to a question raised by Shao et al., that is, for a
non-odd-bipartite hypergraph $G$, that $L(G)$ and $Q(G)$ have the same spectra
can not imply they have the same $H$-spectra. | | Source: | arXiv, 1510.2178 | | Services: | Forum | Review | PDF | Favorites | |
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