18 research outputs found
Removing Twins in Graphs to Break Symmetries
Determining vertex subsets are known tools to provide information about automorphism groups of graphs and, consequently about symmetries of graphs. In this paper, we provide both lower and upper bounds of the minimum size of such vertex subsets, called the determining number of the graph. These bounds, which are performed for arbitrary graphs, allow us to compute the determining number in two different graph families such are cographs and unit interval graphs
Using twins and scaling to construct cospectral graphs for the normalized Laplacian
The spectrum of the normalized Laplacian matrix cannot determine the number
of edges in a graph, however finding constructions of cospectral graphs with
differing number of edges has been elusive. In this paper we use basic
properties of twins and scaling to show how to construct such graphs. We also
give examples of families of graphs which are cospectral with a subgraph for
the normalized Laplacian matrix
Disjoint induced subgraphs of the same order and size
For a graph , let be the largest integer for which there exist
two vertex-disjoint induced subgraphs of each on vertices, both
inducing the same number of edges. We prove that for
every graph on vertices. This answers a question of Caro and Yuster.Comment: 25 pages, improved presentation, fixed misprints, European Journal of
Combinatoric
Twins Vertices in Hypergraphs
Twin vertices in graphs correspond to vertices sharing the same neighborhood. We propose an extension to hypergraphs of the concept of twin vertices. For this we give two characterizations of twin vertices in hypergraphs, a first one in term of clone vertices (the concept of clone has been introduced in [16]) and a second one in term of committees (introduced in [6]). Finally we give an algorithm to aknowledge a set as committee and two algorithms to compute clone-twin vertices classes ans committee-twin vertices classes
Beyond the ego network: The effect of distant connections on node anonymity
Ensuring privacy of individuals is of paramount importance to social network
analysis research. Previous work assessed anonymity in a network based on the
non-uniqueness of a node's ego network. In this work, we show that this
approach does not adequately account for the strong de-anonymizing effect of
distant connections. We first propose the use of d-k-anonymity, a novel measure
that takes knowledge up to distance d of a considered node into account.
Second, we introduce anonymity-cascade, which exploits the so-called
infectiousness of uniqueness: mere information about being connected to another
unique node can make a given node uniquely identifiable. These two approaches,
together with relevant "twin node" processing steps in the underlying graph
structure, offer practitioners flexible solutions, tunable in precision and
computation time. This enables the assessment of anonymity in large-scale
networks with up to millions of nodes and edges. Experiments on graph models
and a wide range of real-world networks show drastic decreases in anonymity
when connections at distance 2 are considered. Moreover, extending the
knowledge beyond the ego network with just one extra link often already
decreases overall anonymity by over 50%. These findings have important
implications for privacy-aware sharing of sensitive network data
Strong Cospectrality and Twin Vertices in Weighted Graphs
We explore algebraic and spectral properties of weighted graphs containing
twin vertices that are useful in quantum state transfer. We extend the notion
of adjacency strong cospectrality to arbitrary Hermitian matrices, with focus
on the generalized adjacency matrix and the generalized normalized adjacency
matrix. We then determine necessary and sufficient conditions such that a pair
of twin vertices in a weighted graph exhibits strong cospectrality with respect
to the above-mentioned matrices. We also generalize known results about
equitable and almost equitable partitions, and use these to determine which
joins of the form , where is either the complete or empty graph,
exhibit strong cospectrality.Comment: 25 pages, 6 figure
Sedentariness in quantum walks
We present a relaxation of the concept of a sedentary family of graphs
introduced by Godsil [Linear Algebra Appl. 614:356-375, 2021] and provide
sufficient conditions for a given vertex in a graph to exhibit sedentariness.
We show that a vertex with at least two twins (vertices that share the same
neighbours) is sedentary. We also prove that there are infinitely many graphs
containing strongly cospectral vertices that are sedentary, which reveals that,
even though strong cospectrality is a necessary condition for pretty good state
transfer, there are strongly cospectral vertices which resist high probability
state transfer to other vertices. Moreover, we derive results about
sedentariness in products of graphs which allow us to construct new sedentary
families, such as Cartesian powers of complete graphs and stars.Comment: 26 pages, 3 figure
Symmetry Parameters of Two-Generator Circulant Graphs
The derived graph of a voltage graph consisting of a single vertex and two
loops of different voltages is a circulant graph with two generators. We
characterize the automorphism groups of connected, two-generator circulant
graphs, and give their determining and distinguishing number, and when
relevant, their cost of 2-distinguishing. We do the same for the subdivisions
of connected, two-generator circulant graphs obtained by replacing one loop in
the voltage graph with a directed cycle