Regularity in Weighted Graphs a Symmetric Function Approach

This work describes how the class of k-regular multigraphs with edge multiplicities from a finite set can be expressed using symmetric species results of Mendez. Consequently, the generating functions can be computed systematically using the scalar product of symmetric functions. This gives conditions on when the classes are D-finite using criteria of Gessel, and a potential route to asymptotic enumeration formulas

On the size of two families of unlabeled bipartite graphs

Let Bu(n,r) denote the set of unlabeled bipartite graphs whose edges connect a set of n vertices with a set of r vertices. In this paper, we provide exact formulas for |Bu(2,r)| and |Bu(3,r)| using Polya's Counting Theorem. Extending these results to n≥4 involves solving a set of complex recurrences and remains open. In particular, the number of recurrences that must be solved to compute |Bu(n,r)| is given by the number of partitions of n that is known to increase exponentially with n by Ramanujan-Hardy-Rademacher's asymptotic formula. © 2017 Kalasalingam University

On The Number of Unlabeled Bipartite Graphs

This paper describes a result that has been obtained in joint work with Abdullah Atmaca of Bilkent University, Ankara, TurkeyLet $I$ and $O$ denote two sets of vertices, where $I\cap O =\Phi$, $|I| = n$, $|O| = r$, and $B_u(n,r)$ denote the set of unlabeled graphs whose edges connect vertices in $I$ and $O$. It is shown that the following two-sided equality holds. $\displaystyle \frac{\binom{r+2^{n}-1}{r}}{n!} \le |B_u(n,r)| \le 2\frac{\binom{r+2^{n}-1}{r}}{n!}