It　is　well-known　that　a　digraph　has　a　spanning　tree　if　and　only　if　its　graph　Laplacian　(matrix)　has　a　simple　zero　eigenvalue　and　all　the　other　nonzero　eigenvalues　have　positive　real　parts.　Different　from　the　approach　of　matrix　and　graph　theory,　this　paper　proposes　a　novel　proof　based　on　a　LaSalle　invariance　principle.　Firstly,　it　is　shown　that　the　max-min　function　considered　in　present　literature　is　a　weak　Lyapunov　function　for　the　error　dynamic　based　on　the　coordinate　transformation　with　any　convex　combination　of　the　agents’　states.　Thus,　the　steady-state　behavior　can　be　characterized　by　the　spanning　tree　condition.　Secondly,　the　proof　relates　directly　the　steady-state　studies　to　the　graph　property.　Such　an　approach　has　a　potential　to　extend　similar　results　to　more　complex　systems　such　as　nonlinear　systems　or　time-varying　systems,　and　thus　provides　a　new　point　of　view　in　studying　consensus　problems　of　multi-agent　systems.　Both　of　them　may　provide　a　new　point　of　view　in　studying　consensus　problems.　©　2023,　International　Frequency　Sensor　Association　(IFSA).　All　rights　reserved.