Some identities for enumerators of circulant graphs

A group induced four-circulant construction for self-dual codes and new extremal binary self-dual codes

We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F_2 + uF_2 and F_4 + uF_4; using groups of order 3, 7, 9, 13, and 15. Through these constructions and their extensions, we find binary self-dual codes of lengths 32, 40, 56, 64, 68 and 80, all of which are extremal or optimal. In particular, we find five new self-dual codes of parameters [56, 28, 10], twenty-three extremal binary self-dual codes of length 68 with new weight enumerators and fifteen new self-dual codes of paramete…

New extremal binary self-dual codes from F_4 + uF_4-lifts of quadratic double circulant codes over F_4

In this work, quadratic double and quadratic bordered double circulant constructions are applied to F_4 + uF_4 as well as F_4, as a result of which extremal binary self-dual codes of length 56 and 64 are obtained. The binary extension theorems as well as the ring extension version are used to obtain 7 extremal self-dual binary codes of length 58, 24 extremal self-dual binary codes of length 66 and 29 extremal self-dual binary codes of length 68, all with new weight enumerators, updating the list of all the known extremal self-dual codes in the literature.

Binary extremal self-dual codes of length $60$ and related codes

We give a classification of four-circulant singly even self-dual $[60,30,d]$ codes for $d=10$ and $12$. These codes are used to construct extremal singly even self-dual $[60,30,12]$ codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual $[60,30,12]$ codes, we also construct extremal singly even self-dual $[58,29,10]$ codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain si…

Constructive and analytic enumeration of circulant graphs with $p^3$ vertices; $p=3,5$

Two methods, structural (constructive) and multiplier (analytical), of exact enumeration of undirected and directed circulant graphs of orders 27 and 125 are elaborated and represented in detail here together with intermediate and final numerical data. The first method is based on the known useful classification of circulant graphs in terms of $S$-rings and results in exhaustive listing (with the use of COCO and GAP) of all corresponding $S$-rings of the indicated orders. The latter method is conducted in the framework of a general approach developed earlier for counting circulant graphs of pr…

第二类欧拉数的恒等式：通过格点计数

An identity for second Eulerian numbers via lattice-point counting

利用 Ehrhart 理论，给出 Bernoulli 数与第二类欧拉数之间两个恒等式的简单概念性证明。

Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model)

We prove some variants of the exponential formula and apply them to the multivariate Tutte polynomials (also known as Potts-model partition functions) of graphs. We also prove some further identities for the multivariate Tutte polynomial, which generalize an identity for counting connected graphs found by Riordan, Nijenhuis, Wilf and Kreweras and in more general form by Leroux and Gessel, and an identity for the inversion enumerator of trees found by Mallows, Riordan and Kreweras. Finally, we prove a generalization of Mobius inversion on the partition lattice.