[For the entire collection see Zbl 0688.00008.]
The author is concerned with two different subjects, which have been developed separately. One is a combinatorial problem in algebraic graph theory, and the other is on arithmetic of discrete subgroups of \(p\)-adic groups and their representations. First he proposes two problems on the number of reduced closed paths of length \(l\) of a finite graph \(X\), which is not a tree, and gives some results. On the other hand, he proves the analogue of Ihara’s zeta function \(Z_{\Gamma}(u)\), which is defined for a discrete subgroup \(\Gamma\) of a \(p\)-adic semisimple algebraic group \({\mathbb G}\), describes the spectral decomposition in \(L^ 2({\mathbb G}/\Gamma)\), of those components which have \({\mathbb B}\)-fixed vectors (\({\mathbb B}\) is an Iwahori subgroup of \({\mathbb G})\). This result is proved based on the fact that \(Z_{\Gamma}(u)\) is related to the zeta functions of finite graphs.