For a random walk on a network, the mean first-passage time from a node i to another node j chosen stochastically according to the equilibrium distribution of Markov chain representing the random walk is called Kemeny constant, which is closely related to the navigability on the network. Thus, the configuration of a network that provides optimal or suboptimal navigation efficiency is a question of interest. It has been proved that complete graphs have the exact minimum Kemeny constant over all graphs. In this paper, by using another method we first prove that complete graphs are the optimal networks with a minimum Kemeny constant, which grows linearly with the network size. Then, we study the Kemeny constant of a class of sparse networks that exhibit remarkable scale-free and fractal features as observed in many real-life networks, which cannot be described by complete graphs. To this end, we determine the closed-form solutions to all eigenvalues and their degeneracies of the networks. Employing these eigenvalues, we derive the exact solution to the Kemeny constant, which also behaves linearly with the network size for some particular cases of networks. We further use the eigenvalue spectra to determine the number of spanning trees in the networks under consideration, which is in complete agreement with previously reported results. Our work demonstrates that scale-free and fractal properties are favorable for efficient navigation, which could be considered when designing networks with high navigation efficiency. Kemeny constant, defined as the mean firstpassage time from a node i to a node j selected randomly according to the equilibrium distribution of Markov chain, is a fundamental quantity for a random walk, since it is a useful indicator characterizing the efficiency of navigation on networks. Here we prove that among all networks, complete graph is the optimal network having the least Kemeny constant, which is consistent with the previous result obtained by a different method. Then, we study the Kemeny constant of a class of sparse scale-free fractal networks, which are ubiquitous in real-life systems. By using the renormalization group technique, we derive the explicit formulas for all eigenvalues and their multiplicities of the networks, based on which we determine the closed-form solution to the Kemeny constant. We show that for some particular cases of networks the leading scaling of Kemeny constant displays the same behavior as that of complete graph, indicating that for networks with a) Electronic mail: zhangzz@fudan.edu.cn; http://homepage.fudan.edu.cn/˜zhangzz/ many nodes, navigation performance similar to that of complete graph can be obtained by networks with many few links. Finally, to show the validity of the eigenvalues and their degeneracies, we also use them to count spanning trees in the networks being studied, and recover previously reported results. Our work is helpful for the structure design of networks where navigation is very efficient.

-The eigenvalue spectrum of the transition matrix of a network encodes important information about its structural and dynamical properties. We study the transition matrix of a family of fractal scale-free networks and analytically determine all the eigenvalues and their degeneracies. We then use these eigenvalues to evaluate the closed-form solution to the eigentime for random walks on the networks under consideration. Through the connection between the spectrum of transition matrix and the number of spanning trees, we corroborate the obtained eigenvalues and their multiplicities.