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Register a new userOn the Spectra of Simplicial Rook Graphs
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The emph{simplicial rook graph} SR(d,n) is the graph whose vertices are the lattice points in the $n$th dilate of the standard simplex in $mathbb{R}^d$, with two vertices adjacent if they differ in exactly two coordinates. We prove that the adjacency and Laplacian matrices of SR(3,n) have integral spectrum for every $n$. The proof proceeds by calculating an explicit eigenbasis. We conjecture that SR(d,n) is integral for all $d$ and $n$, and present evidence in support of this conjecture. For $n<binom{d}{2}$, the evidence indicates that the smallest eigenvalue of the adjacency matrix is $-n$, and that the corresponding eigenspace has dimension given by the Mahonian numbers, which enumerate permutations by number of
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The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of $d$-cycles, $d$-hypertrees and $d$-hypercuts are, respectively, $(d+1)$, $d$, and $(n-d-1)$-vertex-connected. It is also shown that the facet graph of a $d$-cycle cannot be split into more than $s$ connected components by removing at most $s$ vertices. In addition, the paper discusses various related issues, as well as an extension to cell-complexes.
We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers $h$ and $k$ such that $1le hle kle frac{n}{2}$, the Laplacian spectrum of $F_h(G)$ is contained in the Laplacian spectrum of $F_k(G)$. We also show that the double odd graphs and doubled Johnson graphs can be obtained as token graphs of the complete graph $K_n$ and the star $S_{n}=K_{1,n-1}$, respectively. Besides, we obtain a relationship between the spectra of the $k$-token graph of $G$ and the $k$-token graph of its complement $overline{G}$. This generalizes a well-known property for Laplacian eigenvalues of graphs to token graphs. Finally, the double odd graphs and doubled Johnson graphs provide two infinite families, together with some others, in which the algebraic connectivities of the original graph and its token graph coincide. Moreover, we conjecture that this is the case for any graph $G$ and its token graph.
The rook monoid $R_n$ is the finite monoid whose elements are the 0-1 matrices with at most one nonzero entry in each row and column. The group of invertible elements of $R_n$ is isomorphic to the symmetric group $S_n$. The natural extension to $R_n$ of the Bruhat-Chevalley ordering on the symmetric group is defined in cite{Renner86}. In this paper, we find an efficient, combinatorial description of the Bruhat-Chevalley ordering on $R_n$. We also give a useful, combinatorial formula for the length function on $R_n$.
We study the symmetry properties of the spectra of normalized Laplacians on signed graphs. We find a new machinery that generates symmetric spectra for signed graphs, which includes bipartiteness of unsigned graphs as a special case. Moreover, we prove a fundamental connection between the symmetry of the spectrum and the existence of damped two-periodic solutions for the discrete-time heat equation on the graph.
In this paper we answer a question posed by R. Stanley in his collection of Bijection Proof Problems (Problem 240). We present a bijective proof for the enumeration of walks of length $k$ a chess rook can move along on an $mtimes n$ board starting and ending on the same square.
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