386 research outputs found
Sufficient conditions for super k-restricted edge connectivity in graphs of diameter 2
AbstractFor a connected graph G=(V,E), an edge set S⊆E is a k-restricted edge cut if G−S is disconnected and every component of G−S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G)=min{|[X,X¯]|:|X|=k,G[X]is connected}. G is λk-optimal if λk(G)=ξk(G). Moreover, G is super-λk if every minimum k-restricted edge cut of G isolates one connected subgraph of order k. In this paper, we prove that if |NG(u)∩NG(v)|≥2k−1 for all pairs u, v of nonadjacent vertices, then G is λk-optimal; and if |NG(u)∩NG(v)|≥2k for all pairs u, v of nonadjacent vertices, then G is either super-λk or in a special class of graphs. In addition, for k-isoperimetric edge connectivity, which is closely related with the concept of k-restricted edge connectivity, we show similar results
Coboundary expanders
We describe a natural topological generalization of edge expansion for graphs
to regular CW complexes and prove that this property holds with high
probability for certain random complexes.Comment: Version 2: significant rewrite. 18 pages, title changed, and main
theorem extended to more general random complexe
Random walks on dynamic graphs: Mixing times, hitting times, and return probabilities
We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion properties which allows us to capture the progress the random walk makes through t-step probabilities.
We apply our framework to dynamically changing graphs, where the set of vertices is fixed while the set of edges changes in each round. For random walks on dynamic connected graphs for which the stationary distribution does not change over time, we show that their behaviour is in a certain sense similar to static graphs.
For example, we show that the mixing and hitting times of any sequence of d-regular connected graphs is O(n^2), generalising a well-known result for static graphs. We also provide refined bounds depending on the isoperimetric dimension of the graph, matching again known results for static graphs. Finally, we investigate properties of random walks on dynamic graphs that are not always connected: we relate their convergence to stationarity to the spectral properties of an average of transition matrices and provide some examples that demonstrate strong discrepancies between static and dynamic graphs
On the k-restricted edge-connectivity of matched sum graphs
A matched sum graph M of two graphs and of the same order n is obtained by adding to the union (or sum) of and a set M of n independent edges which join vertices in V () to vertices in V (). When and are isomorphic, M is just a permutation graph. In this work we derive
bounds for the k-restricted edge connectivity λ(k) of matched sum graphs M for 2 ≤ k ≤ 5, and present some sufficient conditions for the optimality of λ(k)(M).Peer Reviewe
Minimally 3-restricted edge connected graphs
AbstractFor a connected graph G=(V,E), an edge set S⊂E is a 3-restricted edge cut if G−S is disconnected and every component of G−S has order at least three. The cardinality of a minimum 3-restricted edge cut of G is the 3-restricted edge connectivity of G, denoted by λ3(G). A graph G is called minimally 3-restricted edge connected if λ3(G−e)<λ3(G) for each edge e∈E. A graph G is λ3-optimal if λ3(G)=ξ3(G), where ξ3(G)=max{ω(U):U⊂V(G),G[U] is connected,|U|=3}, ω(U) is the number of edges between U and V∖U, and G[U] is the subgraph of G induced by vertex set U. We show in this paper that a minimally 3-restricted edge connected graph is always λ3-optimal except the 3-cube
On the Spectral Gap of a Quantum Graph
We consider the problem of finding universal bounds of "isoperimetric" or
"isodiametric" type on the spectral gap of the Laplacian on a metric graph with
natural boundary conditions at the vertices, in terms of various analytical and
combinatorial properties of the graph: its total length, diameter, number of
vertices and number of edges. We investigate which combinations of parameters
are necessary to obtain non-trivial upper and lower bounds and obtain a number
of sharp estimates in terms of these parameters. We also show that, in contrast
to the Laplacian matrix on a combinatorial graph, no bound depending only on
the diameter is possible. As a special case of our results on metric graphs, we
deduce estimates for the normalised Laplacian matrix on combinatorial graphs
which, surprisingly, are sometimes sharper than the ones obtained by purely
combinatorial methods in the graph theoretical literature
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