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Abstract
Let G be a simple connected graph with adjacency matrix A. The communicabilityGpq between two nodes p and q of the graph is defined as the pqentry of G=exp(A). We prove here that ξp,q=(Gpp+Gqq2Gpq)1/2 is a Euclidean distance and give expressions for it in paths, cycles, stars and complete graphs with n nodes. The sum of all communicabilitydistances in a graph is introduced as a new graph invariant ϒ(G). We compare this index with the Wiener and Kirchhoff indices of graphs and conjecture about the graphs with maximum and minimum values of this index.
Original language  English 

Pages (fromto)  43174328 
Number of pages  12 
Journal  Linear Algebra and its Applications 
Volume  436 
Issue number  11 
DOIs  
Publication status  Published  1 Jun 2012 
Keywords
 matrix functions
 Euclidean distance
 graph spectrum
 graph distance
 communicability
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Projects
 1 Finished

Mathematics of Large Technological Evolving Networks (MOLTEN)
Higham, D. & Estrada, E.
EPSRC (Engineering and Physical Sciences Research Council)
24/01/11 → 31/03/13
Project: Research