Simple-homotopy for simplicial and CW complexes is a special kind of topological homotopy constructed by elementary collapses and expansions. In this paper we introduce graph homotopy for graphs and Graham homotopy for hypergraphs, and study the relation between these homotopies and the simplehomotopy for simplicial complexes. The graph homotopy is useful to describe topological properties of discretized geometric figures, while the Graham homotopy is essential to characterize acyclic hypergraphs and acyclic relational database schemes.
The sum of distances between all vertex pairs in a connected graph is known as the Wiener index. It is an early index which correlates well with many physico-chemical properties of organic compounds and as such has been well studied over the last quarter of a century. A q-analogue of this index, termed the Wiener polynomial by Hosoya but also known today as the Hosoya polynomial, extends this concept by trying to capture the complete distribution of distances in the graph.Mathematicians have studied several operators on a connected graph in which we see a subdivision of the edges. In this work, we show how the Wiener index of a graph changes with these operations, and extend the results to Wiener polynomials.
A triangle $\{a(n,k)\}_{0\le k\le n}$ of nonnegative numbers is LC-positive
if for each $r$, the sequence of polynomials $\sum_{k=r}^{n}a(n,k)q^k$ is
$q$-log-concave. It is double LC-positive if both triangles $\{a(n,k)\}$ and
$\{a(n,n-k)\}$ are LC-positive. We show that if $\{a(n,k)\}$ is LC-positive
then the log-concavity of the sequence $\{x_k\}$ implies that of the sequence
$\{z_n\}$ defined by $z_n=\sum_{k=0}^{n}a(n,k)x_k$, and if $\{a(n,k)\}$ is
double LC-positive then the log-concavity of sequences $\{x_k\}$ and $\{y_k\}$
implies that of the sequence $\{z_n\}$ defined by
$z_n=\sum_{k=0}^{n}a(n,k)x_ky_{n-k}$. Examples of double LC-positive triangles
include the constant triangle and the Pascal triangle. We also give a
generalization of a result of Liggett that is used to prove a conjecture of
Pemantle on characteristics of negative dependence.Comment: 16 page
Let $T$ be a weighted tree. The weight of a subtree $T_1$ of $T$ is defined
as the product of weights of vertices and edges of $T_1$. We obtain a
linear-time algorithm to count the sum of weights of subtrees of $T$. As
applications, we characterize the tree with the diameter at least $d$, which
has the maximum number of subtrees, and we characterize the tree with the
maximum degree at least $\Delta$, which has the minimum number of subtrees.Comment: 20 pages, 11 figure
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