Related papers: Two kinds of hook length formulas for complete $m$…
A few years ago, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes, both straight and shifted. The formula involves a sum over objects called \emph{excited diagrams}, and the term corresponding to each…
We introduce monoidal width as a measure of the difficulty of decomposing morphisms in monoidal categories. For graphs, we show that monoidal width and two variations capture existing notions, namely branch width, tree width and path width.…
Let $C_{m}$ be a cycle with length $m.$ The $k$-uniform hypercycle with length $m$ obtained by adding $k-2$ new vertices in every edge of $C_{m},$ denoted by $C_{m,k}.$ In this paper, we obtain some trace formulas of uniform hypercycles…
Linear codes with a few weights are very important in coding theory and have attracted a lot of attention. In this paper, we present a construction of $q$-ary linear codes from trace and norm functions over finite fields. The weight…
Let $k\geq2$ be an integer. A $k$-tree is a tree with maximum degree at most $k$. In this paper, we give a closure result on spanning $k$-trees of graphs with given minimum degree. Let $\delta\geq1$ be an integer, and $G$ be a connected…
Over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees or leaf containing subtrees are studied. Here are some of the main results:\ (1)\, Sharp upper bound on the total number…
Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 [JT07]. However, for graphs of tree-width larger than 2, no bound better than NL is known. In this paper, we…
Let $p(m)$ (respectively, $q(m)$) be the maximum number $k$ such that any tree with $m$ edges can be transformed by contracting edges (respectively, by removing vertices) into a caterpillar with $k$ edges. We derive closed-form expressions…
We give exact formulas for the transmission (i.e. the sum of all distances between vertices) of perfect trees and rooted powers of (connected finite) graphs.
Two kinds of evolving trees are considered here: the exponential trees, where subsequent nodes are linked to old nodes without any preference, and the Barab\'asi--Albert scale-free networks, where the probability of linking to a node is…
Bennett, Iosevich and Taylor proved that compact subsets of ${\Bbb R}^d$, $d \ge 2$, of Hausdorff dimensions greater than $\frac{d+1}{2}$ contain chains of arbitrary length with gaps in a non-trivial interval. In this paper we generalize…
Shrock and Wu have given numerical values for the exponential growth rate of the number of spanning trees in Euclidean lattices. We give a new technique for numerical evaluation that gives much more precise values, together with rigorous…
We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions: \frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}(h_u^2 - i^2) = \frac{1}{2(r+1)^2}…
In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in…
Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities that quantify the extent of the "strictness" of the 1-negative type inequalities for finite metric trees. These…
For any integer $k\geq 2$, a spanning $k$-ended tree is a spanning tree with at most $k$ leaves. In this paper, we provide a tight spectral radius condition for the existence of a spanning $k$-ended tree in $t$-connected graphs, which…
This paper investigates the relationships between closed and mixed string amplitudes at the tree level in string theory. Through the analytic continuation of complex variables, we establish a factorization of closed string amplitudes into…
We apply a recent duality theorem for tangles in abstract separation systems to derive tangle-type duality theorems for width-parameters in graphs and matroids. We further derive a duality theorem for the existence of clusters in large data…
In this paper we study a variation of the accessibility percolation model, this is also motivated by evolutionary biology and evolutionary computation. Consider a tree whose vertices are labeled with random numbers. We study the probability…
Recently F\'eray, Goulden and Lascoux gave a proof of a new hook summation formula for unordered increasing trees by means of a generalization of the Pr\"ufer code for labelled trees and posed the problem of finding a bijection between…