Let λKn be the complete λ-fold multigraph of order *n*. Have (G,H) be a λ-fold multigraph pair of order *n* such that:

1. There are no isolated vertices

2. G ≢ H

3. E(G) U E(H) = λKn

In this paper, we found the number of multigraph pairs of orders 2 and 3 of index λ. Additionally, we were able to find partial results for order 4 of index λ, as well as some relations of multigraphs based on structures of subgraphs of *K4, *involving counting and coloring.

This paper experimentally verifies that an asymptotic result on the size of the image for Icart's hash function provided by Fouque and Tibouchi is true for small primes less than 2^{19} and for all curves of conductor less than or equal to 100. Combined with Fouque and Tibouchi's asymptotic result, this proves that the coverage of Icart's hash function is a 5/8 fraction of the points (with some error).

For a graph *G* and a positive integer *a*, define *app _{a}*(