Reliability evaluation of multiprocessor systems is of great significance to the design and maintenance of these systems. As two generalizations of traditional edge connectivity, extra edge connectivity and component edge connectivity are two important parameters to evaluate the fault-tolerant capability of multiprocessor systems. Fast identifying the extra edge connectivity and the component edge connectivity of high order remains a scientific problem for many useful multiprocessor systems. In this article, we determine the $h$ -extra edge connectivity of the 3-ary $n$ -cube $Q_n^3$ for $h\in [1, \frac{3^n-1}{2}]$ . Specifically, we divide the interval $[1, \frac{3^n-1}{2}]$ into some subintervals and characterize the monotonicity of $\lambda _h(Q_n^3)$ in these subintervals and then deduce a recursive closed formula of $\lambda _h(Q_n^3)$ . Based on this formula, an efficient algorithm with complexity $O(\log _3\,N)$ is designed to determine the exact values of $h$ -extra edge connectivity of the 3-ary $n$ -cube $Q_n^3$ for $h\in [1, \frac{3^n-1}{2}]$ completely. Moreover, we also determine the $g$ -component edge connectivity of the 3-ary $n$ -cube $Q_n^3(n\geq 6$ ) for $1\leq g\leq 3^{\lceil \frac{n}{2}\rceil }$ .

Connectivity is an important measure to explore the fault tolerance and reliability of the network structure based on a graph model. Let $G=(V,E)$ be a connected graph. A r-component cut of G is a set S of vertices, G−S has at least r components. The r-component connectivity $c{\kappa }_r\left(G\right)$ of G is the size of the smallest r-component cut. The r-component edge connectivity $c{\lambda }_r\left(G\right)$ can be defined similarly. In this paper, we determine the r-component connectivity of Cayley graphs generated by transposition trees $c{\kappa }_{r+1}\left({\mathcal{T}}_n\right)=r(n-1)-r(r+1)/2+1$ for $r=1,2,\dots ,n-1$ and $n\ge 3$; we also obtain the r-component edge connectivity of Cayley graphs generated by transposition trees ${\mathcal{T}}_n$ for small r and the upper and lower bounds of $c{\lambda }_{r+1}\left({\mathcal{T}}_n\right)$ for $r=1,2,\dots ,n-1$ and $n\ge 3$.

Reliability evaluation of interconnection networks is of significant importance to the design and maintenance of interconnection networks. The extra connectivity is an important parameter for the reliability evaluation of interconnection networks and is a generalization of the traditional connectivity. Let |$g\geq 0$| be an integer and |$G$| be a connected graph; the |$g$|-extra connectivity of |$G$| is the minimum cardinality of a set of vertices in |$G$|⁠, if it exists, whose removal disconnects |$G$| and leaves every component with more than |$g$| vertices. Determining the |$g$|-extra connectivity is still an unsolved problem in many interconnection networks. Let |$n$|⁠, |$k$| be positive integers. Let |$Q_{n,k}\, (1 \leq k \leq n-1)$| denote the |$(n, k)$|-enhanced hypercube. In this paper, we determine the |$g$|-extra connectivity of |$Q_{n,k}$| is |$(n+1)(g+1)-\frac{g(g+3)}{2}$| for |$0\leq g\leq n-k-1$|⁠, |$4\leq k\leq n-5\,(n\geq 9)$|⁠. Some previous results in [Zhang, M. and Zhou, J. (2015) On g-extra connectivity of folded hypercubes. Theor. Comput. Sci., 593, 146–153.] and [Sabir, E., Mamut, A. and Vumar, E. (2019) The extra connectivity of the enhanced hypercubes. Theor. Comput. Sci., 799, 22–31.] are extended.

Assessment on the reliability of interconnection networks plays a significance role in designing and maintaining interconnection networks. The conditional $h$ -edge connectivity $\lambda ^{h}$ , the $h$ -average degree edge connectivity $\overline{\lambda ^{h}}$ , and the $g$ -extra edge connectivity $\lambda _{g}$ are three considerable indicators for assessment of the reliability of interconnection networks, which can maximally improve the real fault tolerability of interconnection networks. In this article, we obtain the relationship between $\lambda ^{h}$ (respectively, $\overline{\lambda ^{h}}$ ) and $\lambda _{g}$ of graphs. Applying this newly established relationship, we obtain $\lambda ^{h}$ (respectively, $\overline{\lambda ^{h}}$ ) of some famous interconnection networks, including 3-ary $n$ -cubes, augmented cubes, and enhanced hypercubes. That is, we show that $\begin{aligned} \lambda ^{h}(\overline{\lambda ^{h}})(Q_{n}^{3})\!=\! \left\lbrace \begin{array}{@{}ll@{}}(2n-h)3^{\frac{h}{2}}, \ \ h\ \text{is even, } 0\leq h\leq 2n-2;\\ (4n \!-\! 2h)3^{\frac{h-1}{2}}, \ h\ \text{is odd, } 0\leq h\leq 2n \!-\! 3; \end{array} \right. \end{aligned}$ when $k\ne 2$ , $ \lambda ^{h}(\overline{\lambda ^{h}})(Q_{n,k})= \left\lbrace \begin{array}{@{}ll@{}}(n+1-h)2^{h}, \ 0\leq h\leq n-k;\\ (n+1-h)2^{h-1}, \ n\!-\!k \!+\! 2 \leq h\leq n; \end{array} \right. $ when $k= 2$ , $ \lambda ^{h}(\overline{\lambda ^{h}})(Q_{n,k})= \left\lbrace \begin{array}{@{}ll@{}}(n+1-h)2^{h}, \ 0\leq h\leq n-3;\\ 2^{n-1}, \ h= n-2, n-1; \end{array} \right. $ and $\lambda ^{h}(AQ_{n})=\overline{\lambda ^{h}}(AQ_{n})=(n-\frac{h+1}{2})2^{\frac{h+3}{2}}$ for $h$ is odd within the range $1\leq h\leq 2n-3$ . In particular, these results extend the previous results in [ J. Supercomput. 2022, 5: 6739-6751] and [ Inform. Process. Lett. 2008, 106: 59-63] and also solve positively a conjecture presented by Shinde and Borse [ J. Interconnect. Netw. 2020, 20: 2050013].