103,155 research outputs found
Extremal Properties of Complex Networks
We describe the structure of connected graphs with the minimum and maximum
average distance, radius, diameter, betweenness centrality, efficiency and
resistance distance, given their order and size. We find tight bounds on these
graph qualities for any arbitrary number of nodes and edges and analytically
derive the form and properties of such networks
Bounds on the radius and status of graphs
Two classical concepts of centrality in a graph are the median and the
center. The connected notions of the status and the radius of a graph seem to
be in no relation. In this paper, however, we show a clear connection of both
concepts, as they obtain their minimum and maximum values at the same type of
tree graphs. Trees with fixed maximum degree and extremum radius and status,
resp., are characterized. The bounds on radius and status can be transferred to
general connected graphs via spanning trees.
A new method of proof allows not only to regain results of Lin et al. on
graphs with extremum status, but it allows also to prove analogous results on
graphs with extremum radius
Spectral radius of finite and infinite planar graphs and of graphs of bounded genus
It is well known that the spectral radius of a tree whose maximum degree is
cannot exceed . In this paper we derive similar bounds for
arbitrary planar graphs and for graphs of bounded genus. It is proved that a
the spectral radius of a planar graph of maximum vertex degree
satisfies . This result is
best possible up to the additive constant--we construct an (infinite) planar
graph of maximum degree , whose spectral radius is . This
generalizes and improves several previous results and solves an open problem
proposed by Tom Hayes. Similar bounds are derived for graphs of bounded genus.
For every , these bounds can be improved by excluding as a
subgraph. In particular, the upper bound is strengthened for 5-connected
graphs. All our results hold for finite as well as for infinite graphs.
At the end we enhance the graph decomposition method introduced in the first
part of the paper and apply it to tessellations of the hyperbolic plane. We
derive bounds on the spectral radius that are close to the true value, and even
in the simplest case of regular tessellations of type we derive an
essential improvement over known results, obtaining exact estimates in the
first order term and non-trivial estimates for the second order asymptotics
Algorithms and almost tight results for 3-colorability of small diameter graphs.
The 3-coloring problem is well known to be NP-complete. It is also well known that it remains NP-complete when the input is restricted to graphs with diameter 4. Moreover, assuming the Exponential Time Hypothesis (ETH), 3-coloring cannot be solved in time 2o(n) on graphs with n vertices and diameter at most 4. In spite of extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of this problem on graphs with small diameter, i.e. with diameter at most 2, or at most 3, has been an open problem. In this paper we investigate graphs with small diameter. For graphs with diameter at most 2, we provide the first subexponential algorithm for 3-coloring, with complexity 2O(nlognā). Furthermore we extend the notion of an articulation vertex to that of an articulation neighborhood, and we provide a polynomial algorithm for 3-coloring on graphs with diameter 2 that have at least one articulation neighborhood. For graphs with diameter at most 3, we establish the complexity of 3-coloring by proving for every Īµā[0,1) that 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with n vertices and minimum degree Ī“=Ī(nĪµ). Moreover, assuming ETH, we use three different amplification techniques of our hardness results, in order to obtain for every Īµā[0,1) subexponential asymptotic lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree Ī“=Ī(nĪµ). Finally, we provide a 3-coloring algorithm with running time 2O(min{Ī“Ī, nĪ“logĪ“}) for arbitrary graphs with diameter 3, where n is the number of vertices and Ī“ (resp. Ī) is the minimum (resp. maximum) degree of the input graph. To the best of our knowledge, this is the first subexponential algorithm for graphs with Ī“=Ļ(1) and for graphs with Ī“=O(1) and Ī=o(n). Due to the above lower bounds of the complexity of 3-coloring, the running time of this algorithm is asymptotically almost tight when the minimum degree of the input graph is Ī“=Ī(nĪµ), where Īµā[12,1), as its time complexity is 2O(nĪ“logĪ“)=2O(n1āĪµlogn) and the corresponding lower bound states that there is no 2o(n1āĪµ)-time algorithm
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