305 research outputs found
Algorithm and Complexity for a Network Assortativity Measure
We show that finding a graph realization with the minimum Randi\'c index for
a given degree sequence is solvable in polynomial time by formulating the
problem as a minimum weight perfect b-matching problem. However, the
realization found via this reduction is not guaranteed to be connected.
Approximating the minimum weight b-matching problem subject to a connectivity
constraint is shown to be NP-Hard. For instances in which the optimal solution
to the minimum Randi\'c index problem is not connected, we describe a heuristic
to connect the graph using pairwise edge exchanges that preserves the degree
sequence. In our computational experiments, the heuristic performs well and the
Randi\'c index of the realization after our heuristic is within 3% of the
unconstrained optimal value on average. Although we focus on minimizing the
Randi\'c index, our results extend to maximizing the Randi\'c index as well.
Applications of the Randi\'c index to synchronization of neuronal networks
controlling respiration in mammals and to normalizing cortical thickness
networks in diagnosing individuals with dementia are provided.Comment: Added additional section on application
On the extremal properties of the average eccentricity
The eccentricity of a vertex is the maximum distance from it to another
vertex and the average eccentricity of a graph is the mean value
of eccentricities of all vertices of . The average eccentricity is deeply
connected with a topological descriptor called the eccentric connectivity
index, defined as a sum of products of vertex degrees and eccentricities. In
this paper we analyze extremal properties of the average eccentricity,
introducing two graph transformations that increase or decrease .
Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX,
about the average eccentricity and other graph parameters (the clique number,
the Randi\' c index and the independence number), refute one AutoGraphiX
conjecture about the average eccentricity and the minimum vertex degree and
correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure
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