Related papers: Action graphs and Catalan numbers
This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…
A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that…
We prove that for each $k\ge0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, so that the expected number of vertices of degree $k$ is asymptotically $d_k n$, and moreover…
Let $G$ be a finite simple graph on the vertex set $V(G) = \{x_1, \ldots, x_n\}$ and $I(G) \subset K[V(G)]$ its edge ideal, where $K[V(G)]$ is the polynomial ring in $x_1, \ldots, x_n$ over a field $K$ with each ${\rm deg} x_i = 1$ and…
In this paper, we consider the problem of reconstructing a directed graph using path queries. In this query model of learning, a graph is hidden from the learner, and the learner can access information about it with path queries. For a…
We say that a vertex $v$ in a connected graph $G$ is decisive if the numbers of walks from $v$ of each length determine the graph $G$ rooted at $v$ up to isomorphism among all connected rooted graphs with the same number of vertices. On the…
We prove that finding a $k$-edge induced subgraph is fixed-parameter tractable, thereby answering an open problem of Leizhen Cai. Our algorithm is based on several combinatorial observations, Gauss' famous \emph{Eureka} theorem [Andrews,…
This paper will show when a rooted path tree of a finite directed rooted graph has only finitely many orbits under the action of its undirected automorphism group (i.e. when it is cocompact). This will allow us to specify which trees are…
Let $c_n$ denote the number of nodes at a distance $n$ from the root of a rooted tree. A criterion for proving the rationality and computing the rational generating function of the sequence $\{c_n\}$ is described. This criterion is applied…
We define the (random) $k$-cut number of a rooted graph to model the difficulty of the destruction of a resilient network. The process is as the cut model of Meir and Moon except now a node must be cut $k$ times before it is destroyed. The…
We investigate certain nonassociative binary operations that satisfy a four-parameter generalization of the associative law. From this we obtain variations of the ubiquitous Catalan numbers and connections to many interesting combinatorial…
We give a generating function for the number of graphs with given numerical properties and prescribed weighted number of connected components. As an application, we give a generating function for the number of bipartite graphs of given…
Consider two planar graphs which are subject to edge insertions and deletions. We show that whether the two graphs are isomorphic can be maintained with first-order logic formulas and auxiliary data of polynomial size. This places the…
We introduce a graph-theoretical interpretation of an induced action of Aut$(\Gamma)$ in the discrete de Rham cohomology of a finite graph $\Gamma$. This action produces a splitting of Aut$(\Gamma)$ that depends on the cycles of $\Gamma$.…
A successive vertex ordering of a graph is a linear ordering of its vertices in which every vertex except the first has at least one neighbour appearing earlier. Such orderings arise naturally in incremental growth and…
The presented material is devoted to the equivalent conversion from the vertex graphs to the edge graphs. We suggest that the proved theorems solve the problem of the isomorphism of graphs, the problem of the graph's enumeration with the…
(DRAFT VERSION) In this article we present a proof of the famous Kirchoff's Matrix-Tree theorem, which relates the number of spanning trees in a connected graph with the cofactors (and eigenvalues) of its combinatorial Laplacian matrix.…
A \emph{queue layout} of a graph consists of a \emph{linear order} of its vertices and a partition of its edges into \emph{queues}, so that no two independent edges of the same queue are nested. The \emph{queue number} of a graph is the…
Graphs, and sequences of growing graphs, can be used to specify the architecture of mathematical models in many fields including machine learning and computational science. Here we define structured graph "lineages" (ordered by level…
In the first part, we introduce a notion a degree of edge-colorings of bicubic plane graphs and proves some local formula of the graded number of colorings. In the second part, we give a new proof of a result of Fisk saying that any two…