KPPY 58

Let $G$ be a graph. For a real number $c$, a {\it $c$-sum flow} of $G$ is an assignment of non-zero real numbers to the edges of $G$ such that the sum of values of all edges incident with each vertex is $c$. Let $k$ be a natural number. A {\it $c$-sum $k$-flow} is a $c$-sum flow with values from the set \mbox{$\{\pm 1,\ldots ,\pm(k-1)\}$}. In this talk, we present known results on $c$-sum $k$-flows of graphs and propose several conjectures. For a hypergraph $H$, a {\it $0$-sum flow}, means a nowhere-zero real vector in the null space of the incidence matrix of $H$. Here, we state some results on $0$-sum flows of hypergraphs.

In this talk we use some linear algebraic tools and graph factorization methods to obtain some results in zero-sum flows and their generalizations. Let $A$ be an abelian group and $A^*=A\setminus \{0\}$. For a subset $S \subseteq A$, a map $\phi : E(G) \rightarrow S$ is called an {\it $S$-flow}. For a given $S$-flow of $G$, and every $v\in V(G)$, define $s(v) =\sum_{uv \in E(G)} \phi(uv)$. It is shown that if $G$ is a $2$-edge connected bipartite graph with two parts $X=\{x_1,\ldots,x_r\}$ and $Y=\{y_1,\ldots,y_s\}$ and $c_1,\ldots,c_r,d_1,\ldots,d_s$ are arbitrary integers, then there exists a $\mathbb{Z}^*$-flow of $G$ such that $s(x_i)=c_i$ and $s(y_j)=d_j$, for $1 \leq i \leq r$, $1 \leq j \leq s$ if and only if $\sum_{i=1}^{r}c_i=\sum_{j=1}^{s}d_j$.

Recently, I and Dr. Hanchul Park have computed the $i$-th (rational) Betti number of the real toric variety associated to a graph $G$. It can be calculated by a purely combinatorial method and is called the $i$-th $a$-number of $G$. In the talk, I am willing to introduce recent works on the topic from the viewpoint of combinatorics and poset theory.

The Erdős-Szekeres Theorem states that in any sufficiently large set of points in the plane, there exists a subset consisting of n points in convex position. We analyse what happens when we replace points by lines in this statement.

We study the symmetric operators introduced by Haglund, Morse and Zabrocki which generate Hall-Littlewood polynomials indexed by compositions. They considered these operators to study the combinatorics of the characters of the space of diagonal harmonics. We construct a combinatorial way of applying these symmetric operators to various symmetric bases and use them to prove theorems of Haglund, Morse and Zabrocki combinatorially. Also, using the combinatorial construction of applying the symmetric operators to Schur functions, we can derive a new recursion of charge statistic.

For Lie algebra $\rm{sl}(2)$, the invariants vectors are known as Temperley Lieb algebras. Since the dimension of these invariants spaces is Catalan number, it endows with some combinatorial descriptions. We will explain how the Westbury diagram can be constructed for other Lie algebras including $\rm{sl}(3)$, $\rm{sl}(4)$ and more.