The ShieldsHarary numbers of Km,n K m , n for continuous concave cost functions vanishing at 1
It is shown that the ShieldsHarary index of vulnerability of the complete bipartite graph Km,n K m , n , with respect to the cost function f(x)=1-x f ( x ) = 1 - x , 0?x?1 0 ? x ? 1 , is m , if n ? m + 2 m , and 1 n + 1 ? ( n + m ) 2 4 ? , if m ? n < m + 2 m . It follows that the ShieldsHarary number of Km,n K m , n with respect to any concave continuous cost function f  on [0,1] [ 0 , 1 ] satisfying f(1)=0 f ( 1 ) = 0 is mf(0) mf ( 0 ) , if n ? m + 2 m , and between 1 n + 1 ? ( n + m ) 2 4 ? f ( 0 ) and mf(0) mf ( 0 ) , if m ? n < m + 2 m .
shieldsharary numbers concave cost function network vulnerability