13 research outputs found
Graph isomorphism and volumes of convex bodies
We show that a nontrivial graph isomorphism problem of two undirected graphs,
and more generally, the permutation similarity of two given
matrices, is equivalent to equalities of volumes of the induced three convex
bounded polytopes intersected with a given sequence of balls, centered at the
origin with radii , where is an increasing
sequence converging to . These polytopes are characterized by
inequalities in at most variables. The existence of fpras for computing
volumes of convex bodies gives rise to a semi-frpas of order at
most to find if given two undirected graphs are isomorphic.Comment: 9 page
The Graph Isomorphism Problem and approximate categories
It is unknown whether two graphs can be tested for isomorphism in polynomial
time. A classical approach to the Graph Isomorphism Problem is the
d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can
distinguish many pairs of graphs, but the pairs of non-isomorphic graphs
constructed by Cai, Furer and Immerman it cannot distinguish. If d is fixed,
then the WL-algorithm runs in polynomial time. We will formulate the Graph
Isomorphism Problem as an Orbit Problem: Given a representation V of an
algebraic group G and two elements v_1,v_2 in V, decide whether v_1 and v_2 lie
in the same G-orbit. Then we attack the Orbit Problem by constructing certain
approximate categories C_d(V), d=1,2,3,... whose objects include the elements
of V. We show that v_1 and v_2 are not in the same orbit by showing that they
are not isomorphic in the category C_d(V) for some d. For every d this gives us
an algorithm for isomorphism testing. We will show that the WL-algorithms
reduce to our algorithms, but that our algorithms cannot be reduced to the
WL-algorithms. Unlike the Weisfeiler-Lehman algorithm, our algorithm can
distinguish the Cai-Furer-Immerman graphs in polynomial time.Comment: 29 page
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Descriptive complexity of graph spectra.
Two graphs are cospectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are cospectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting first-order logic are cospectral, and this is not the case with , nor with any number of variables if we exclude counting quantifiers. We also show that the class of graphs that are determined by their spectra is definable in partial fixed-point logic with counting. We relate these properties to other algebraic and combinatorial problems.OZ was supported by CONACyT-Mexico Grant 384665, SS was supported by EPSRC and The Royal Society
On the Expressive Power of Linear Algebra on Graphs
Most graph query languages are rooted in logic. By contrast, in this paper we consider graph query languages rooted in linear algebra. More specifically, we consider MATLANG, a matrix query language recently introduced, in which some basic linear algebra functionality is supported. We investigate the problem of characterising equivalence of graphs, represented by their adjacency matrices, for various fragments of MATLANG. A complete picture is painted of the impact of the linear algebra operations in MATLANG on their ability to distinguish graphs
On symmetric association schemes and associated quotient-polynomial graphs
Let denote an undirected, connected, regular graph with vertex set , adjacency matrix , and distinct eigenvalues. Let denote the subalgebra of generated by . We refer to as the adjacency algebra of . In this paper we investigate algebraic and combinatorial structure of for which the adjacency algebra is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) has a standard basis ; (ii) for every vertex there exists identical distance-faithful intersection diagram of with cells; (iii) the graph is quotient-polynomial; and (iv) if we pick then has distinct eigenvalues if and only if . We describe the combinatorial structure of quotient-polynomial graphs with diameter and distinct eigenvalues. As a consequence of the techniques used in the paper, some simple algorithms allow us to decide whether is distance-regular or not and, more generally, which distance- matrices are polynomial in , giving also these polynomials.This research has been partially supported by AGAUR from the Catalan
Government under project 2017SGR1087 and by MICINN from the Spanish Government under
project PGC2018-095471-B-I00. The second author acknowledges the financial support from the
Slovenian Research Agency (research program P1-0285 and research project J1-1695).Peer ReviewedPostprint (published version
Permutation group approach to association schemes
AbstractWe survey the modern theory of schemes (coherent configurations). The main attention is paid to the schurity problem and the separability problem. Several applications of schemes to constructing polynomial-time algorithms, in particular, graph isomorphism tests, are discussed