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Paper IPM / M / 17632  


Abstract:  
Let G be a finite simple noncomplete connected graph on {1, . . . , n}
and Îº(G) â¥ 1 its vertex connectivity. Let f(G) denote the number of free vertices
of G and diam(G) the diameter of G. Being motivated by the computation of
the depth of the binomial edge ideal of G, the possible sequences (n, q, f, d) of
integers for which there is a finite simple noncomplete connected graph G on
{1, . . . , n} with q = Îº(G), f = f(G), d = diam(G) satisfying f + d = n + 2 â q will
be determined. Furthermore, finite simple noncomplete connected graphs G on
{1, . . . , n} satisfying f(G) + diam(G) = n + 2 â Îº(G) will be classified.
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