Related papers: The complexity of the $q$-analog of the $n$-cube

The number of spanning trees of a graph $G$ is called the {\em complexity} of $G$ and is denoted $c(G)$. Let C(n) denote the {\em (binary) hypercube} of dimension $n$. A classical result in enumerative combinatorics (based on explicit…

We define a q-analog of the adjacency matrix of the n-cube, determine its eigenvalues and write down a canonical eigenbasis. We give a weighted count of the number of rooted spanning trees in the q-analog of the n-cube. Remarks on the…

We give two combinatorial proofs of an elegant product formula for the number of spanning trees of the $n$-dimensional hypercube. The first proof is based on the assertion that if one chooses a uniformly random rooted spanning tree of the…

Let $L(G)$ denote the maximum number of leaves in any spanning tree of a connected graph $G$. We show the (known) result that for the $n$-cube $Q_n$, $L(Q_n) \sim 2^n = |V(Q_n)|$ as $n\rightarrow \infty$. Examining this more carefully,…

We give a direct combinatorial proof of the known fact that the 3-cube has 384 spanning trees, using an "edge slide" operation on spanning trees. This gives an answer in the case n=3 to a question implicitly raised by Stanley. Our argument…

Using the special value at $u=1$ of the Artin-Ihara $L$-function, we give a short proof of the count of the number of spanning trees in the $n$-cube.

Given an elliptic curve C, we study here $N_k = #C(F_{q^k})$, the number of points of C over the finite field F_{q^k}. This sequence of numbers, as k runs over positive integers, has numerous remarkable properties of a combinatorial flavor…

Just how many different connected shapes result from slicing a cube along some of its edges and unfolding it into the plane? In this article we answer this question by viewing the cube both as a surface and as a graph of vertices and edges.…

Let $Q$ be an affine quiver of type $A_2^{(1)}$. We explicitly construct the cluster multiplication formulas for the quantum cluster algebra of $Q$ with principal coefficients. As applications, we obtain: (1)\ an exact expression for every…

The signature of a spanning tree $T$ of the $n$-cube $Q_n$ is the $n$-tuple $\mathrm{sig}(T)=(a_1,a_2,\dots,a_n)$ such that $a_i$ is the number of edges of $T$ in the $i$th direction. We characterise the $n$-tuples that can occur as the…

We construct explicity the automorphism group of the folded hypercube $FQ_n$ of dimension $n>3$, as a semidirect product of $N$ by $M$, where $N$ is isomorphic to the Abelian group $Z_2^n$, and $M$ is isomorphic to $Sym(n+1)$, the symmetric…

There are different concepts regarding to tree decomposition of a graph $G$. For the Hypercube $Q_n$, these concepts have been shown to have many applications. But some diverse papers on this subject make it difficult to follow what is…

Kontsevich conjectured that the number f(G,q) of zeros over the finite field with q elements of a certain polynomial connected with the spanning trees of a graph G is polynomial function of q. We have been unable to settle Kontsevich's…

We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is…

The complexity of a graph is the number of its labeled spanning trees. In this work complexity is studied in settings that admit regular graphs. An exact formula is established linking complexity of the complement of a regular graph to…

We prove a new q-analogue of Nicomachus's Theorem about the sum of cubes and some related results.

In the present paper we suggest a simple approach for counting Jacobian group of the $Y$-graph $Y(n; k, l, m).$ In the case $Y(n; 1, 1, 1)$ the structure of the Jacobian group will be find explicitly. Also, we obtain a closed formula for…

We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial…

We give explicit positive combinatorial interpretations for the plethysm coefficients $\langle s_\mu[s_\nu], s_\lambda\rangle$, when $\lambda$ has at most two rows, as counting certain marked trees. In the special case $\mu=(n)$, this also…

We consider all spanning trees of a complete simple graph $\Gamma$ on $n$ vertices that contain a given $m-$forest $F$. We show that the number of such spanning trees, $\tau(F)$, doesn't depend on the structure of $F$ and is completely…