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We introduce a new matroid (graph) invariant, the arboricity polynomial. Given a matroid, the arboricity polynomial enumerates the number of covers of the ground set by disjoint independent sets. We establish the polynomiality of the…

We propose an end-to-end deep learning learning model for graph classification and representation learning that is invariant to permutation of the nodes of the input graphs. We address the challenge of learning a fixed size graph…

Polynomials on stranded graphs are higher dimensional generalization of Tutte and Bollob\'as-Riordan polynomials [Math. Ann. 323 (2002), 81-96]. Here, we deepen the analysis of the polynomial invariant defined on rank 3 weakly-colored…

In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted $U$-polynomial for every $k$; by this we mean that the polynomials agree on terms with degree at most $k+1$. The main tool for this…

Given a connected undirected graph $G$, a spanning tree is a subgraph $T$ of $G$ such that $V(T) = V(G)$ and $T$ is a tree. A collection of $\ell$ spanning trees $T_1,\ldots,T_\ell$ is pairwise $k$-diverse if for every $i \neq j$, $|E(T_i)…

Phylogenetic networks are a generalisation of phylogenetic trees that allow for more complex evolutionary histories that include hybridisation-like processes. It is of considerable interest whether a network can be considered `tree-like' or…

The Tutte polynomial is a generalization of the chromatic polynomial of graph colorings. Here we present an extension called the rooted Tutte polynomial, which is defined on a graph where one or more vertices are colored with prescribed…

This paper contains the first knot polynomials which can distinguish the orientations of classical knots and which make no excplicit use of the knot group. But they make extensive use of the meridian and of the longitude in a geometric way.…

In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials. We explore some of the Tutte polynomial's many properties and applications and we use the Tutte polynomial to…

We study Thurston equivalence classes of quadratic post-critically finite branched coverings. For these maps, we introduce and study invariant spanning trees. We give a computational procedure for searching for invariant spanning trees.…

The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by…

The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex…

Phylogenetic networks generalize phylogenetic trees, and have been introduced in order to describe evolution in the case of transfer of genetic material between coexisting species. There are many classes of phylogenetic networks, which can…

Rooted phylogenetic networks are used to describe evolutionary histories that contain non-treelike evolutionary events such as hybridization and horizontal gene transfer. In some cases, such histories can be described by a phylogenetic…

Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its arithmetic Tutte polynomial.…

A graph is a $k$-leaf power of a tree $T$ if its vertices are leaves of $T$ and two vertices are adjacent in $T$ if and only if their distance in $T$ is at most $k$. Then $T$ is a $k$-leaf root of $G$. This notion was introduced by…

The graph isomorphism problem is theoretically interesting and also has many practical applications. The best known classical algorithms for graph isomorphism all run in time super-polynomial in the size of the graph in the worst case. An…

We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in…

We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs…

Richard Stanley defined the chromatic symmetric function $X_G$ of a graph $G$ and asked whether there are non-isomorphic trees $T$ and $U$ with $X_T=X_U$. We study variants of the chromatic symmetric function for rooted graphs, where we…