10,019 research outputs found
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four
Spectral preorder and perturbations of discrete weighted graphs
In this article, we introduce a geometric and a spectral preorder relation on
the class of weighted graphs with a magnetic potential. The first preorder is
expressed through the existence of a graph homomorphism respecting the magnetic
potential and fulfilling certain inequalities for the weights. The second
preorder refers to the spectrum of the associated Laplacian of the magnetic
weighted graph. These relations give a quantitative control of the effect of
elementary and composite perturbations of the graph (deleting edges,
contracting vertices, etc.) on the spectrum of the corresponding Laplacians,
generalising interlacing of eigenvalues.
We give several applications of the preorders: we show how to classify graphs
according to these preorders and we prove the stability of certain eigenvalues
in graphs with a maximal d-clique. Moreover, we show the monotonicity of the
eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic
Cheeger constants with respect to the geometric preorder. Finally, we prove a
refined procedure to detect spectral gaps in the spectrum of an infinite
covering graph.Comment: 26 pages; 8 figure
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