Let \(H_{G,v_0}\) be the Hardy operator defined on the finite graph \(G=(V,E)\) and \(v_0\in V\) . We obtain the \(\ell ^p\) -norm estimates for the operator \(H_{G,v_0}\) in the corresponding strong-type (p, p) for \(0<p<\infty \) and weak-type (p, p) for \(0<p\le \infty \) . That is, (i) we obtain the upper and lower bounds for the \(\ell ^p\) -norm of a general graph G in the range \(0<p\le 1\) and characterize a special kind of connected graph by the value of the operator norm \(\Vert H_{G,v_0}\Vert _{\ell ^{p}\rightarrow \ell ^{p}}\) ; (ii) we obtain sharp constants for \(\Vert H_{G,v_0}\Vert _{\ell ^{p}\rightarrow \ell ^{p}}\) in the range \(0<p<\infty \) when G is the complete graph \(K_n\) , the star graph \(S_n\) and the path \(P_n\) with \(n\ge 2\) ; (iii) we compute the weak-type norm \(\Vert H_{G,v_0}\Vert _{\ell ^{p}\rightarrow \ell ^{p,\infty }}\) with \(0< p\le \infty \) . Finally, we calculate the p-variation inequalities for the operator \(H_{G,v_0}\) when \(G=K_{n}\) , \(G=S_{n}\) and \(G=P_{n}\) with \(0<p\le \infty \) .
