[Paper Review] The Connection between the Number of Realizations for Degree Sequences and Majorization

Annabell Berger|arXiv (Cornell University)|Dec 21, 2012

Markov Chains and Monte Carlo Methods38 references3 citations

TL;DR

This paper establishes a novel connection between majorization and the number of graph, bipartite, and digraph realizations for given degree sequences. It proves that if a degree sequence $ S' $ majorizes another sequence $ S $, then $ S $ has strictly more realizations than $ S' $, with constant or minconvex sequences maximizing realization counts under fixed $ n $ and $ m $.

ABSTRACT

The \emph{graph realization problem} is to find for given nonnegative integers $a_1,\dots,a_n$ a simple graph (no loops or multiple edges) such that each vertex $v_i$ has degree $a_i.$ Given pairs of nonnegative integers $(a_1,b_1),\dots,(a_n,b_n),$ (i) the \emph{bipartite realization problem} ask whether there is a bipartite graph (no loops or multiple edges) such that vectors $(a_1,...,a_n)$ and $(b_1,...,b_n)$ correspond to the lists of degrees in the two partite sets, (ii) the \emph{digraph realization problem} is to find a digraph (no loops or multiple arcs) such that each vertex $v_i$ has indegree $a_i$ and outdegree $b_i.$\\ The classic literature provides characterizations for the existence of such realizations that are strongly related to the concept of majorization. Aigner and Triesch (1994) extended this approach to a more general result for graphs, leading to an efficient realization algorithm and a short and simple proof for the Erdős-Gallai Theorem. We extend this approach to the bipartite realization problem and the digraph realization problem.\\ Our main result is the connection between majorization and the number of realizations for a degree list in all three problems. We show: if degree list $S'$ majorizes $S$ in a certain sense, then $S$ possesses more realizations than $S'.$ We prove that constant lists possess the largest number of realizations for fixed $n$ and a fixed number of arcs $m$ when $n$ divides $m.$ So-called \emph{minconvex lists} for graphs and bipartite graphs or \emph{opposed minconvex lists} for digraphs maximize the number of realizations when $n$ does not divide $m$.

Motivation & Objective

• To extend the majorization framework from undirected graphs to bipartite and directed graphs.
• To characterize how the number of realizations of a degree sequence relates to its majorization order.
• To identify the degree sequences that maximize the number of realizations under fixed $ n $ and $ m $.
• To provide a unified theoretical foundation for counting realizations using majorization and transfer paths.

Proposed method

• Uses the classical majorization relation $ \prec $ on integer vectors to compare degree sequences.
• Applies unit transfer operations to transform sequences and analyze changes in realization counts.
• Leverages known characterization theorems (Erdős-Gallai, Gale-Ryser, Fulkerson) as foundational constraints.
• Employs symmetric digraph representations to link digraphic lists to graphic lists via bijective mapping.
• Uses inductive arguments on transfer paths to prove monotonicity of realization counts under majorization.
• Introduces the concept of minconvex and opposed minconvex lists as extremal cases for maximizing realizations.

Experimental results

Research questions

• RQ1 How does majorization relate to the number of realizations of a degree sequence in graph, bipartite, and digraph settings?
• RQ2 Does a sequence that majorizes another always have fewer realizations?
• RQ3 Which degree sequences maximize the number of realizations for fixed $ n $ and $ m $?
• RQ4 Can the number of realizations be bounded using majorization and transfer paths?
• RQ5 What structural properties define the extremal sequences (e.g., constant or minconvex lists) in terms of realization count?

Key findings

• If $ a \prec a' $ and $ a \neq a' $, then the number of graph realizations of $ a $ is strictly greater than that of $ a' $, i.e., $ N_3(a) > N_3(a') $.
• Constant degree sequences maximize the number of realizations when $ n $ divides $ m $, the total number of edges.
• When $ n $ does not divide $ m $, minconvex lists for graphs and bipartite graphs, or opposed minconvex lists for digraphs, maximize the number of realizations.
• The number of realizations is strictly decreasing along any transfer path from a majorized sequence to its dominator.
• For symmetric digraphic lists, the number of realizations corresponds bijectively to the number of graphic realizations, preserving the majorization-based ordering.
• The minconvex list $ \alpha $ for a given $ m $ and $ n $ satisfies $ \alpha \prec a $ for any other nonincreasing graphic list $ a $, and thus $ N_3(\alpha) \geq N_3(a) $.