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The determination of mean first-passage time (MFPT) for random walks innetworks is a theoretical challenge, and is a topic of considerable recentinterest within the physics community. In this paper, according to the knownconnections between MFPT, effective resistance, and the eigenvalues of graphLaplacian, we first study analytically the MFPT between all node pairs of aclass of growing treelike networks, which we term deterministic uniformrecursive trees (DURTs), since one of its particular cases is a deterministicversion of the famous uniform recursive tree. The interesting quantity isdetermined exactly through the recursive relation of the Laplacian spectraobtained from the special construction of DURTs. The analytical result showsthat the MFPT between all couples of nodes in DURTs varies as $N \ln N$ forlarge networks with node number $N$. Second, we study trapping on a particularnetwork of DURTs, focusing on a special case with the immobile trap positionedat a node having largest degree. We determine exactly the average trapping time(ATT) that is defined as the average of FPT from all nodes to the trap. Incontrast to the scaling of the MFPT, the leading behavior of ATT is a linearfunction of $N$. Interestingly, we show that the behavior for ATT of thetrapping problem is related to the trapping location, which is in comparisonwith the phenomenon of trapping on fractal T-graph although both networksexhibit treestructure. Finally, we believe that the methods could open the wayto exactly calculate the MFPT and ATT in a wide range of deterministic media.

Explicit determination of mean first-passage time for random walks on deterministic uniform recursive trees