Related papers: Enumerative properties of Ferrers graphs
In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bipartite graph from its adjacency…
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…
Inspired by a concept in comparative genomics, we investigate properties of randomly chosen members of G_1(m,n,t), the set of bipartite graphs with $m$ left vertices, n right vertices, t edges, and each vertex of degree at least one. We…
In a recent work on the bipartite Erd\H{o}s-R\'{e}nyi graph, Do et al. (2023) established upper bounds on the number of connected labeled bipartite graphs with a fixed surplus. We use some recent encodings of bipartite random graphs in…
Some important properties of the chromatic polynomial also hold for any polynomial set map satisfying p_S(x+y)=\sum_{T\uplus U=S}p_T(x)p_U(y). Using umbral calculus, we give a formula for the expansion of such a set map in terms of any…
Given a finite set of $2$-edge-coloured graphs $\mathcal F$ and a hereditary property of graphs $\mathcal{P}$, we say that $\mathcal F$ expresses $\mathcal{P}$ if a graph $G$ has the property $\mathcal{P}$ if and only if it admits a…
Using the theory of combinatorial species, we compute the cycle index for bipartite graphs, which we use to count unlabeled bipartite graphs and bipartite blocks.
We present a graph theoretical approach to the configurational statistics of random tree-like objects, such as randomly branching polymers. In particular, for ideal trees we show that Pr\"ufer labelling provides: (i) direct access to the…
F-nomial coefficients encompass among others well-known binomial coefficients or Gaussian coefficients that count subsets of finite set and subspaces of finite vector space respectively. Here, the so called F-cobweb tiling sequences N(a)…
In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic…
We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind,…
In this paper we give a complete characterization of the statistical equivalence classes of CEGs and of staged trees. We are able to show that all graphical representations of the same model share a common polynomial description. Then,…
We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and colored permutations. The corresponding…
We consider the bialgebra of hypergraphs, a generalization of Schmitt's Hopf algebra of graphs, and show it has a cointeracting bialgebra. So one has a double bialgebra in the sense of L. Foissy, who recently proved there is then a unique…
We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs…
Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander…
This work considers new entropy-based proofs of some known, or otherwise refined, combinatorial bounds for bipartite graphs. These include upper bounds on the number of the independent sets, lower bounds on the minimal number of colors in…
We study a $q$-version of the chromatic polynomial of a given graph $G=(V,E)$, namely, \[ \chi_G^\lambda(q,n) \ := \sum_{\substack{\text{proper colorings}\\ c\,:\,V\to[n]}} q^{ \sum_{ v \in V } \lambda_v c(v) }, \] where $\lambda \in…
In this paper, we consider multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously…