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This paper focuses on the time constant for last passage percolation oncomplete graph. Let $G_n=([n],E_n)$ be the complete graph on vertex set$[n]=\{1,2,\ldots,n\}$, and i.i.d. sequence $\{X_e:e\in E_n\}$ be the passagetimes of edges. Denote by $W_n$ the largest passage time among allself-avoiding paths from 1 to $n$. First, it is proved that $W_n/n$ convergesto constant $\mu$, where $\mu$ is called the time constant and coincides withthe essential supremum of $X_e$. Second, when $\mu<\infty$, it is proved thatthe deviation probability $P(W_n/n\leq \mu-x)$ decays as fast as$e^{-\Theta(n^2)}$, and as a corollary, an upper bound for the variance of$W_n$ is obtained. Finally, when $\mu=\infty$, lower and upper bounds for$W_n/n$ are given.

On The Time Constant for Last Passage Percolation on Complete Graph