Combinatorics of symmetric plabic graphs

Classifying pentavalnet symmetric graphs of order $24p$

A graph is said to be symmetric if its automorphism group is transitive on its arcs. A complete classification is given of pentavalent symmetric graphs of order 24p for each prime p. It is shown that a connected pentavalent symmetric graph of order 24p exists if and only if p=2, 3, 5, 11 or 17, and up to isomorphism, there are only eleven such graphs.

Combinatorics of theq-Basis of Symmetric Functions

A basis of symmetric functions, which we denote by q*(X; q, t), was introduced in the work of Ram and King and Wybourne in order to describe the irreducible characters of the Hecke algebras of type A. In this work we give combinatorial descriptions of the expansions of the functions q*(X; q, t) in terms of the classical bases of symmetric functions and apply these results in determining the det...

Analytic Combinatorics of Random Graphs

Analytic combinatorics of connected graphs

We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, i.e. generating function manipulations, we derive the complete asymptotic expansion. keywords. connected graphs, analytic combinatorics, generating functions, asymptotic expan...

Cubic symmetric graphs of orders $36p$ and $36p^{2}$

A graph is textit{symmetric}, if its automorphism group is transitive on the set of its arcs. In this paper, we  classifyall the connected cubic symmetric  graphs of order $36p$  and $36p^{2}$, for each prime $p$, of which the proof depends on the classification of finite simple groups.