Laplacian Distribution and Domination

Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $\gamma(G)$ denote its domination number. We extend the recent result $m_G[0,1) \leq \gamma(G)$, and show that isolate-free graphs also satisfy $\gamma(G) \leq m_G[2,n]$. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of $\gamma(G)$, showing that $\frac{\gamma(G)}{m_G[0,1)} \not\in O(\log n)$. However, $\gamma(G) \leq m_G[2, n] \leq (c + 1) \gamma(G)$ for $c$-cyclic graphs, $c \geq 1$. For trees $T$, $\gamma(T) \leq m_T[2, n] \leq 2 \gamma(G)$

Signless normalized Laplacian for hypergraphs

The spectral theory of the normalized Laplacian for chemical hypergraphs is further investigated. The signless normalized Laplacian is introduced and it is shown that its spectrum for classical hypergraphs coincides with the spectrum of the normalized Laplacian for bipartite chemical hypergraphs. Furthermore, the spectra of special families of hypergraphs are established

On the spectrum of hypergraphs

Here we study the spectral properties of an underlying weighted graph of a non-uniform hypergraph by introducing different connectivity matrices, such as adjacency, Laplacian and normalized Laplacian matrices. We show that different structural properties of a hypergrpah, can be well studied using spectral properties of these matrices. Connectivity of a hypergraph is also investigated by the eigenvalues of these operators. Spectral radii of the same are bounded by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by the eigenvalues of its connectivity matrices. We characterize different properties of a regular hypergraph characterized by the spectrum. Strong (vertex) chromatic number of a hypergraph is bounded by the eigenvalues. Cheeger constant on a hypergraph is defined and we show that it can be bounded by the smallest nontrivial eigenvalues of Laplacian matrix and normalized Laplacian matrix, respectively, of a connected hypergraph. We also show an approach to study random walk on a (non-uniform) hypergraph that can be performed by analyzing the spectrum of transition probability operator which is defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two different ways. We show that if the Laplace operator, $\Delta$, on a hypergraph satisfies a curvature-dimension type inequality $CD (\mathbf{m}, \mathbf{K})$ with $\mathbf{m}>1$ and $\mathbf{K}>0$ then any non-zero eigenvalue of $- \Delta$ can be bounded below by $\frac{ \mathbf{m} \mathbf{K}}{ \mathbf{m} -1 }$. Eigenvalues of a normalized Laplacian operator defined on a connected hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph