Let S_n and A_{n} denote the symmetric and alternating group on the set {1,ldots ,n}, respectively. In this paper we are interested in the second largest eigenvalue lambda _{2}(Gamma ) of the Cayley graph Gamma =mathrm{Cay}(G,H) over G=S_{n} or A_{n} for certain connecting sets H. Let 1<kle n and denote the set of all k-cycles in S_{n} by C(n, k). For H=C(n,n) we prove that lambda _{2}(Gamma )=(n-2)! (when n is even) and lambda _{2}(Gamma )=2(n-3)! (when n is odd). Further, for H=C(n,n-1) we have lambda _{2}(Gamma )=3(n-3)(n-5)! (when n is even) and lambda _{2}(Gamma )=2(n-2)(n-5) ! (when n is odd). The case H=C(n,3) has been considered in Huang and Huang (J Algebraic Combin 50:99–111, 2019). Let 1le r<k<n and let C(n,k;r) subseteq C(n,k) be set of all k-cycles in S_{n} which move all the points in the set {1,2,ldots ,r}. That is to say, g=(i_{1},i_{2},ldots ,i_{k})(i_{k+1})dots (i_{n})in C(n,k;r) if and only if {1,2,ldots ,r}subset {i_{1},i_{2},ldots ,i_{k}}. Our main result concerns lambda _{2}(Gamma ), where Gamma =mathrm{Cay}(G,H) with H=C(n,k;r) with 1le r<k<n when G=S_{n} if k is even and G=A_{n} if k is odd. Here we observe that λ2(Γ)≥(k-2)!n-rk-r1n-r((k-1)(n-k)-(k-r-1)(k-r)n-r-1).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\lambda _{2}(\\Gamma )\\ge (k-2)! {n-r \\atopwithdelims ()k-r} \\frac{1}{n-r} \\big ((k-1)(n-k) - \\frac{(k-r-1)(k-r)}{n-r-1}\\big ). \\end{aligned}$$\\end{document}We prove that this bound is attained in the special case k=r+1 , giving lambda _{2}(Gamma )=r!(n-r-1). The cases with H=C(n,3;1) and H=C(n,3;2) were considered earlier in [6].