[Paper Review] The Ring of Graph Invariants - Graphic Values

Tomi Mikkonen|ArXiv.org|Dec 3, 2007

Computational Geometry and Mesh Generation7 references3 citations

TL;DR

This paper introduces $G$-posets to study the ring of graph invariants, focusing on subgraph counting invariants and their algebraic dependencies. It develops weak and strong notions of 'graphic values' using inequalities derived from small $G$-posets, proves graph reconstruction from invariants is NP-complete under restrictions, and formulates Ramsey numbers as integer polyhedron problems in adjustable dimensions.

ABSTRACT

The ring of graph invariants is spanned by the basic graph invariants which calculate the number of subgraphs isomorphic to a given graph in other graphs. These subgraphs counting invariants are not algebraically independent. In our view the most important problem in graph theory of unlabeled graphs is the problem of determining graphic values of arbitrary sets of graph invariants. This corresponds to explaining the syzygy of the graph invariants when the number of vertices is unbounded. We introduce two methods to explore this complicated structure. Sets of graphs with a small number of vertices impose constraints on larger sets. We describe families of inequalities of graph invariants. These inequalities allow to loop over all values of graph invariants which look like graphic from the small sets point of view. We also develop strong notion of graphic values where the existence of the corresponding graphs is guaranteed once the constraints are satisfied by the basic graph invariants. These constraints are necessary and sufficient for graphs whose local neighborhoods are generated by a finite set of locally connected graphs. The reconstruction of the graph from the basic graph invariants is shown to be NP-complete in this restricted case. Finally we apply these results to formulate the problem of Ramsey numbers as an integer polyhedron problem of moderate and adjustable dimension.

Motivation & Objective

• To understand the algebraic structure of graph invariants by analyzing their syzygies and dependencies.
• To address the fundamental problem of determining which combinations of invariants correspond to actual graphs—i.e., 'graphic values'.
• To develop a framework for distinguishing between weak graphic values (feasible by small $G$-posets) and strong graphic values (guaranteed to exist via constraints).
• To connect graph invariants to Ramsey theory by reformulating Ramsey number problems as integer programming problems.
• To investigate the complexity of reconstructing graphs from their basic invariants, showing it is NP-complete under certain restrictions.

Proposed method

• Uses $G$-posets $\mathcal{E}(n)$ to organize basic graph invariants counting subgraphs isomorphic to a given graph.
• Employs the $E$-transform matrix $E = (e_{ij}) = I(g_j)(g_i)$ to encode subgraph counts between graphs.
• Applies the inverse $B = E^{-1}$ to derive linear relations among invariants, enabling syzygy computation.
• Derives families of inequalities from small $G$-posets ($\mathcal{E}(4)$, $\mathcal{E}(5)$) to define weak graphic values.
• Introduces strong graphic values via necessary and sufficient constraints for graphs with locally finite neighborhood structures.
• Reduces Ramsey number problems to integer polyhedron problems by parameterizing invariants using power sums $\sigma_e^v$.

Experimental results

Research questions

• RQ1 What is the minimal $G$-poset $\mathcal{E}(r)$ required to prove upper bounds on Ramsey numbers $r(k)$?
• RQ2 Can local parameters $z(i)$, $z(i_1,i_2)$, etc., strengthen constraints for computing $r(4)$ or $r(5)$?
• RQ3 Is it possible to derive new lower bounds for Ramsey numbers using the reconstruction theorem and $G$-poset constraints?
• RQ4 How can the structure of syzygies among $\sigma_e^v$ parameters be characterized for general graph invariants?
• RQ5 What is the computational complexity of reconstructing a graph from its basic invariants, and how does it scale with graph size?

Key findings

• The $E$-transform matrix $E$ for $\mathcal{E}(4)$ successfully proves $r(3) \leq 6$, while $\mathcal{E}(5)$ is insufficient for $r(4) \leq 18$.
• Weak graphic values are defined via inequalities from small $G$-posets, but do not guarantee existence of corresponding graphs.
• Strong graphic values are characterized by necessary and sufficient constraints for graphs whose local neighborhoods are generated by finite sets of locally connected graphs.
• Graph reconstruction from basic invariants is NP-complete in the restricted case where local neighborhood structures are finite.
• Ramsey number problems can be reformulated as integer polyhedron problems of moderate and adjustable dimension using $\sigma_e^v$ parameters.
• Syzygy computation for $r(5)$ requires $\mathcal{E}(10)$, and current implementations exceed desktop memory limits, indicating high computational cost.