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Register a new userThe reduced formula of the characteristic polynomial of hypergraphs and the spectrum of hyperpaths
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In this paper, we give a reduced formula of the characteristic polynomial of $k$-uniform hypergraphs with a pendant edge. And the explicit characteristic polynomial and all distinct eigenvalues of $k$-uniform hyperpath are given.
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Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{alpha}(G)=alpha D(G)+(1-alpha)A(G) $$ for any real $alphain [0,1]$. The $A_{alpha}$-characteristic polynomial of $G$ is defined to be $$ det(xI_n-A_{alpha}(G))=sum_jc_{alpha j}(G)x^{n-j}, $$ where $det(*)$ denotes the determinant of $*$, and $I_n$ is the identity matrix of size $n$. The $A_{alpha}$-spectrum of $G$ consists of all roots of the $A_{alpha}$-characteristic polynomial of $G$. A graph $G$ is said to be determined by its $A_{alpha}$-spectrum if all graphs having the same $A_{alpha}$-spectrum as $G$ are isomorphic to $G$. In this paper, we first formulate the first four coefficients $c_{alpha 0}(G)$, $c_{alpha 1}(G)$, $c_{alpha 2}(G)$ and $c_{alpha 3}(G)$ of the $A_{alpha}$-characteristic polynomial of $G$. And then, we observe that $A_{alpha}$-spectra are much efficient for us to distinguish graphs, by enumerating the $A_{alpha}$-characteristic polynomials for all graphs on at most 10 vertices. To verify this observation, we characterize some graphs determined by their $A_{alpha}$-spectra.
In arXiv:1709.07504 Aguiar and Ardila give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on hypergraphs. We give a combinatorial interpretation of this invariant on negative integers which leads to a reciprocity theorem on hypergraphs. Finally, we use this invariant to recover well-known invariants on other combinatorial objects (graphs, simplicial complexes, building sets etc) as well as the associated reciprocity theorems.
Let $mathcal{G}$ be a $k$-uniform hypergraph, $mathcal{L}_{mathcal{G}}$ be its Laplacian tensor. And $beta( mathcal{G})$ denotes the maximum number of linearly independent nonnegative eigenvectors of $mathcal{L}_{mathcal{G}}$ corresponding to the eigenvalue $0$. In this paper, $beta( mathcal{G})$ is called the geometry connectivity of $mathcal{G}$. We show that the number of connected components of $mathcal{G}$ equals the geometry connectivity $beta( mathcal{G})$.
We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial differential, diffusion-like, equations. These equations are valid for matrices of arbitrary size. Their solutions can be given an integral representation that allows for a simple study of their asymptotic behaviors for a broad range of initial conditions.
The determinants of ${pm 1}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of $0$ to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamards maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.
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