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Let Γ n be the symmetric group on { 1 , 2 , … , n } and S be the generating set of Γ n . The corresponding Cayley graph is denoted by Γ n ( S ) . If all elements of S are transpositions, a simple way to depict S is to via a graph, called the transposition generating graph of S , denoted by A ( S ) (or say simply A ), where the vertex set of A is { 1 , 2 , … , n } , there is an edge in A between i and j if and only if the transposition ( i j ) ∈ S , and Γ n ( S ) is called a Cayley graph obtained from a transposition generating graph A . Conditional diagnosability, proposed by Lai et al, is a novel measure of diagnosability that adds the additional condition that any faulty set cannot contain all of the neighbors of any vertex in a system. There are two well-known diagnostic models, PMC model and MM* model. The conditional diagnosability under the PMC (resp. MM*) model of a graph G , denoted by t c P M C ( G ) (resp. t c M M ∗ ( G ) ), is the maximum value of t is the maximum value of t such that G is conditionally t -diagnosable under the PMC (resp. MM*) model. The conditional diagnosability of many well-known multiprocessor systems has been explored. In this paper, by exploring the structural properties of these Cayley graphs, suppose | E ( A ) | is the number of edges in the transposition generating graph A , we obtain the following results: (1) Under the MM* model, t c M M ∗ ( Γ n ( S ) ) = 3 | E ( A ) | − 6 , if A has a triangle; 3 | E ( A ) | − 5 , if A has no triangles and A is not a star. (2) Under the PMC model, t c P M C ( Γ n ( S ) ) = 4 | E ( A ) | − 9 , if A has a triangle; 4 | E ( A ) | − 7 , if A has no triangles and A is not a star. As corollaries, the conditional diagnosability of many kinds of graphs under the MM* model and the PMC model such as Cayley graphs generated by unicyclic graphs, wheel graphs, complete graphs, tree graphs etc. are obtained.

A conditional connectivity is the generalization of the traditional connectivity and can provide more accurate measures regarding the reliability of a large-scale multiprocessor system. The extra edge connectivity and component edge connectivity are two parameters for the reliability evaluation. Although these two measures have attracted considerable attention and been determined for many classes of graphs in recent years, they are established independently. In this paper, we determine the relationship between extra edge connectivities and component edge connectivities of regular networks. Using this relationship, the extra edge connectivity and component edge connectivity are explored for some well-known graphs, including BC graphs, data center networks, augmented k-ary n-cubes, hierarchical star networks, burnt pancake graphs, ( n , k ) -star graphs, etc.

Given a connected graph $G$ and a positive integer $\ell $, the $\ell $-extra (resp. $\ell $-component) edge connectivity of $G$, denoted by $\lambda ^{(\ell )}(G)$ (resp. $\lambda _{\ell }(G)$), is the minimum number of edges whose removal from $G$ results in a disconnected graph so that every component has more than $\ell $ vertices (resp. so that it contains at least $\ell $ components). This naturally generalizes the classical edge connectivity of graphs defined in term of the minimum edge cut. In this paper, we proposed a general approach to derive component (resp. extra) edge connectivity for a connected graph $G$. For a connected graph $G$, let $S$ be a vertex subset of $G$ for $G\in \{\Gamma _{n}(\Delta ),AG_n,S_n^2\}$ such that $|S|=s\leq |V(G)|/2$, $G[S]$ is connected and $|E(S,G-S)|=\min \limits _{U\subseteq V(G)}\{|E(U, G-U)|: |U|=s, G[U]\ \textrm{is connected}\ \}$, then we prove that $\lambda ^{(s-1)}(G)=|E(S,G-S)|$ and $\lambda _{s+1}(G)=|E(S,G-S)|+|E(G[S])|$ for $s=3,4,5$. By exploring the reliability analysis of $AG_n$ and $S_n^2$ based on extra (component) edge faults, we obtain the following results: (i) $\lambda _3(AG_n)-1=\lambda ^{(1)}(AG_n)=4n-10$, $\lambda _4(AG_n)-3=\lambda ^{(2)}(AG_n)=6n-18$ and $\lambda _5(AG_n)-4=\lambda ^{(3)}(AG_n)=8n-24$; (ii) $\lambda _3(S_n^2)-1=\lambda ^{(1)}(S_n^2)=4n-8$, $\lambda _4(S_n^2)-3=\lambda ^{(2)}(S_n^2)=6n-15$ and $\lambda _5(S_n^2)-4=\lambda ^{(3)}(S_n^2)=8n-20$. This general approach maybe applied to many diverse networks.

A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $\min \{\textrm{deg}_G(x), \textrm{deg}_G(y)\}$ internally disjoint (edge disjoint) paths between $x$ and $y$. Motivated by parallel routing in networks with faults, Oh and Chen (resp., Qiao and Yang) proposed the (fault-tolerant) strong Menger (edge) connectivity as follows. A graph $G$ is called $m$-strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$. A graph $G$ is called $m$-conditional strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$ and $\delta (G-F)\geq 2$. In this paper, we consider strong Menger (edge) connectedness of the augmented $k$-ary $n$-cube $AQ_{n,k}$, which is a variant of $k$-ary $n$-cube $Q_n^k$. By exploring the topological proprieties of $AQ_{n,k}$, we show that $AQ_{n,3}$ (resp. $AQ_{n,k}$, $k\geq 4$) is $(4n-9)$-strongly (resp. $(4n-8)$-strongly) Menger connected for $n\geq 4$ (resp. $n\geq 2$) and $AQ_{n,k}$ is $(4n-4)$-strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. Moreover, we obtain that $AQ_{n,k}$ is $(8n-10)$-conditional strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.

The l -component connectivity of a graph G , denoted by c κ l ( G ) , is the minimum number of vertices whose removal from G results in a disconnected graph with at least l components or a graph with fewer than l vertices. This is a natural generalization of the classical connectivity of graphs defined in terms of the minimum vertex-cut. Since this parameter can be used to evaluate the reliability and fault tolerance of a graph G corresponding to a network, determining the exact values of c κ l ( G ) is an important issue on the research topic of networks. However, it has been pointed out in Hsu et al. (2012) that determining l -component connectivity is still unsolved in most interconnection networks even for small l ’s. Let B S n and B P n denote the n -dimensional bubble-sort star graph and the n -dimensional burnt pancake graph, respectively. In this paper, for B S n , we determine the values: c κ 3 ( B S n ) = 4 n − 9 for n ⩾ 3 , and c κ 4 ( B S n ) = 6 n − 16 and c κ 5 ( B S n ) = 8 n − 24 for n ⩾ 4 . Similarly, for B P n , we determine the values: c κ 3 ( B P n ) = 2 n − 1 and c κ 4 ( B P n ) = 3 n − 2 for n ⩾ 4 , and c κ 5 ( B P n ) = 4 n − 4 for n ⩾ 5 .

For an integer [Formula: see text], the [Formula: see text]-component connectivity of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of vertices whose removal from [Formula: see text] results in a disconnected graph with at least [Formula: see text] components or a graph with fewer than [Formula: see text] vertices. This naturally generalizes the classical connectivity of graphs defined in term of the minimum vertex-cut. This kind of connectivity can help us to measure the robustness of the graph corresponding to a network. The hierarchical star networks [Formula: see text], proposed by Shi and Srimani, is a new level interconnection network topology, and uses the star graphs as building blocks. In this paper, by exploring the combinatorial properties and fault-tolerance of [Formula: see text], we study the [Formula: see text]-component connectivity of hierarchical star networks [Formula: see text]. We obtain the results: [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text].

A graph G = ( V , E ) is F-strongly Menger-vertex-connected (resp. Menger-edge-connected), if for a subgraph G − F of G with a conditional faulty vertex (resp. edge) set F ⊆ V (resp. F ⊆ E ), each pair of vertices u and v are connected by min { deg G − F ( u ) , deg G − F ( v ) } internally disjoint (resp. edge-disjoint) fault-free paths in G − F . Where deg G − F ( u ) and deg G − F ( v ) are the degrees of u and v in G − F , respectively. He et al. (2018) [7] introduced some sufficient conditions of F-strongly Menger (edge) connected for k-regular triangle-free graphs. In this paper, we consider the strongly Menger connectedness of two kinds of graphs with triangles: data center network D k , n and ( n , k ) -star graph S n , k . We prove D k , n is ( k − 1 ) -strongly Menger-vertex-connected, ( n + k − 3 ) -strongly Menger-edge-connected of order 1, ( 2 n + 2 k − 8 ) -strongly Menger-edge-connected of order 2, and ( 3 n + 3 k − 15 ) -strongly Menger-edge-connected of order 3. Furthermore, we obtain S n , k is ( k − 2 ) -strongly Menger-vertex-connected, ( n − 3 ) -strongly Menger-edge-connected of order 1, ( 2 n − 8 ) -strongly Menger-edge-connected of order 2, and ( 3 n − 15 ) -strongly Menger-edge-connected of order 3.

Most graphs have this property: after removing a linear number of vertices from a graph, the surviving graph is either connected or consists of a large connected component and small components containing a small number of vertices. This property can be applied to derive fault-tolerance related network parameters: extra edge connectivity and component edge connectivity. Using this general property, we obtained the $h$-extra edge connectivity and $(h+2)$-component edge connectivity of augmented cubes, Cayley graphs generated by transposition trees, complete cubic networks (including hierarchical cubic networks), generalized exchanged hypercubes (including exchanged hypercubes) and dual-cube-like graphs (including dual cubes).

For an integer $\ell \geqslant 2$ , the $\ell $ -component connectivity (resp. $\ell $ -component edge connectivity) of a graph $G$ , denoted by $\kappa _{\ell }(G)$ (resp. $\lambda _{\ell }(G)$ ), is the minimum number of vertices (resp. edges) whose removal from $G$ results in a disconnected graph with at least $\ell $ components. The two parameters naturally generalize the classical connectivity and edge connectivity of graphs defined in term of the minimum vertex-cut and the minimum edge-cut, respectively. The two kinds of connectivities can help us to measure the robustness of the graph corresponding to a network. In this paper, by exploring algebraic and combinatorial properties of $n$ -dimensional balanced hypercubes $BH_n$ , we obtain the $\ell $ -component (edge) connectivity $\kappa _{\ell }(BH_n)$ ( $\lambda _{\ell }(BH_n)$ ). For $\ell $ -component connectivity, we prove that $\kappa _2(BH_n)=\kappa _3(BH_n)=2n$ for $n\geq 2$ , $\kappa _4(BH_n)=\kappa _5(BH_n)=4n-2$ for $n\geq 4$ , $\kappa _6(BH_n)=\kappa _7(BH_n)=6n-6$ for $n\geq 5$ . For $\ell $ -component edge connectivity, we prove that $\lambda _3(BH_n)=4n-1$ , $\lambda _4(BH_n)=6n-2$ for $n\geq 2$ and $\lambda _5(BH_n)=8n-4$ for $n\geq 3$ . Moreover, we also prove $\lambda _\ell (BH_n)\leq 2n(\ell -1)-2\ell +6$ for $4\leq \ell \leq 2n+3$ and the upper bound of $\lambda _\ell (BH_n)$ we obtained is tight for $\ell =4,5$ .

For an integer ℓ ≥ 2, the ℓ-component connectivity of a graph G, denoted by κℓ(G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least ℓ components or a graph with fewer than ℓ vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and a good measure of vulnerability for the graph corresponding to a network. So far, the exact values of ℓ-connectivity are known only for a few classes of networks and small ℓ's. It has been pointed out in component connectivity of the hypercubes, International Journal of Computer Mathematics 89 (2012) 137-145] that determining ℓ-connectivity is still unsolved for most interconnection networks such as alternating group graphs and star graphs. In this paper, by exploring the combinatorial properties and the fault-tolerance of the alternating group graphs AGn and a variation of the star graphs called split-stars Sn2, we study their ℓ-component connectivities. We obtain the following results: 1) κ3(AGn) = 4n - 10 and κ4(AGn) = 6n - 16 for n ≥ 4, and κ5(AGn) = 8n - 24 for n ≥ 5 and 2) κ3(Sn2) = 4n - 8, κ4(Sn2) = 6n - 14, and κ5(Sn2) = 8n - 20 for n ≥ 4.

An interconnection network's diagnosability is an important measure of its self-diagnostic capability. In 2012, Peng et al. proposed a measure for fault diagnosis of the network, namely, the $h$-good-neighbor conditional diagnosability, which requires that every fault-free node has at least $h$ fault-free neighbors. There are two well-known diagnostic models, PMC model and MM* model. The {\it $h$-good-neighbor diagnosability} under the PMC (resp. MM*) model of a graph $G$, denoted by $t_h^{PMC}(G)$ (resp. $t_h^{MM^*}(G)$), is the maximum value of $t$ such that $G$ is $h$-good-neighbor $t$-diagnosable under the PMC (resp. MM*) model. In this paper, we study the $2$-good-neighbor diagnosability of some general $k$-regular $k$-connected graphs $G$ under the PMC model and the MM* model. The main result $t_2^{PMC}(G)=t_2^{MM^*}(G)=g(k-1)-1$ with some acceptable conditions is obtained, where $g$ is the girth of $G$. Furthermore, the following new results under the two models are obtained: $t_2^{PMC}(HS_n)=t_2^{MM^*}(HS_n)=4n-5$ for the hierarchical star network $HS_n$, $t_2^{PMC}(S_n^2)=t_2^{MM^*}(S_n^2)=6n-13$ for the split-star networks $S_n^2$ and $t_2^{PMC}(\Gamma_{n}(\Delta))=t_2^{MM^*}(\Gamma_{n}(\Delta))=6n-16$ for the Cayley graph generated by the $2$-tree $\Gamma_{n}(\Delta)$.

Let Γ n be the symmetric group on { 1 , 2 , … , n } and S be the generating set of Γ n . The corresponding Cayley graph is denoted by Γ n ( S ) . If all elements of S are transpositions, a simple way to depict S is via a graph, called the transposition generating graph of S , denoted by A ( S ) (or say simply A ), where the vertex set of A is { 1 , 2 , … , n } , there is an edge in A between i and j if and only if the transposition ( i j ) ∈ S , and Γ n ( S ) is called a Cayley graph obtained from a transposition generating graph A . In this paper, by exploring and utilizing the structural properties of these Cayley graphs, we obtain that the pessimistic diagnosability of Γ n ( S ) is equal to 2 | E ( A ) | − 2 if A has no triangles or 2 | E ( A ) | − 3 if A has a triangle. As corollaries, the pessimistic diagnosability of many kinds of graphs such as Cayley graphs generated by unicyclic graphs, wheel graphs, complete graphs, and tree graphs is obtained.

A system is t / t -diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free processor mistaken as a faulty one. The pessimistic diagnosability of a system G, denoted by t p ( G ) , is the maximal number of faulty processors so that the system G is t / t -diagnosable. Data centers are critical to the business of companies such as Amazon, Google, Facebook, and Microsoft. Based on data centers, the data center networks D n , k , given in 2008, has many desirable features. In this paper, by exploring the structure of D k , n , we firstly determined the pessimistic diagnosability of D k , n and prove that t p ( D k , n ) = n + 2 k − 2 for k ≥ 2 and n ≥ 2 under the PMC model. Then we prove that D k , n ( k ≥ 0 , n ≥ 2 ) is not edge transitive except the cases of k = 0 and k = 1 , n = 2 .

The hierarchical star networks , proposed by Shi and Srimani, is a new level network topology and uses the star graphs as building blocks. Given a graph G, the h-extra connectivity of G is the mini...

In this paper, we determine the neighbor connectivity κNB of two kinds of Cayley graphs: alternating group networks AN n and star graphs S n ; and give the exact values of edge neighbor connectivity λNB of ANn and Cayley graphs generated by transposition trees Γ n . Those are κNB(AN n ) = n−1, λNB(AN n ) = n−2 and κNB(S n ) = λNB(Γ n ) = n−1.

A system is t / t -diagnosable if, provided the number of faulty processors is bounded by t , all faulty processors can be isolated within a set of size at most t with at most one fault-free processor mistaken as a faulty one. The pessimistic diagnosability of a system G , denoted by t p ( G ) , is the maximal number of faulty processors so that the system G is t / t -diagnosable. The pessimistic diagnosability of alternating group graphs A G n (Tsai, 2015); BC networks (Fan, 2005; Tsai, 2013); the k -ary n -cube networks Q n k , (Wang etźal., 2012); regular graphs including the alternating group networks A N n (Hao etźal., 2016) etc. But most of these results are about networks G with c n ( G ) ź 2 (where c n ( G ) is the maximum number of common neighbors for any two distinct vertices). In this paper, we study the pessimistic diagnosability of three kinds of graphs which are ( n , k ) -arrangement graphs A n , k , ( n , k ) -star graphs S n , k and balanced hypercubes B H n , where c n ( A n , k ) = c n ( S n , k ) = n - k - 1 and c n ( B H n ) = 2 n . We proved that t p ( A n , k ) = ( 2 k - 1 ) ( n - k ) - 1 for n ź k + 2 and k ź 3 , t p ( S n , k ) = n + k - 3 for n ź k + 2 and k ź 3 , and t p ( B H n ) = 2 n for n ź 2 . As corollaries, the pessimistic diagnosability of the known results about A G n and A N n is obtained directly.

Many large-scale multiprocessor or multi-computer systems take interconnection networks as underlying topologies. Fault diagnosis is especially important to identify fault tolerability of such systems. The g-good-neighbor (conditional) diagnosability such that every fault-free node has at least g fault-free neighbors is a novel measure of diagnosability. In this paper, we show that the g-good-neighbor diagnosability of the ( n , k ) -star graph S n , k under the PMC model ( 2 ź k ź n - 1 and 1 ź g ź n - k ) and the comparison model ( 2 ź k ź n - 1 and 2 ź g ź n - k ) is n + g ( k - 1 ) - 1 , respectively. In addition, we derive that 1-good-neighbor diagnosability of S n , k under the comparison model is n + k - 2 for 3 ź k ź n - 1 and n ź 4 . As a supplement, we also derive that the g-good-neighbor diagnosability of the ( n , 1 ) -star graph S n , 1 ( 1 ź g ź ź n / 2 ź - 1 and n ź 4 ) under the PMC model and the comparison model is ź n / 2 ź - 1 , respectively. We explore fault diagnosability of multiprocessor systems based on combinatorial network theory.We establish the g-good-neighbor diagnosability of the ( n , k ) -star graph S n , k under the PMC model.We establish the g-good-neighbor diagnosability of the ( n , k ) -star graph S n , k under the comparison model.

The neighbor connectivity of a graph $G$ is the least number of vertices such that removing their closed neighborhoods from $G$ results in a graph that is disconnected, complete or empty. If a~graph is used to model the topology of an interconnection network, this means that the failure of a network node causes failures of all its neighbors. We completely determine the neighbor connectivity of $k$-ary $n$-cubes for all $n\ge1$ and $k\ge2$., Comment: 11 pages, 1 figure

A system is t/t-diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free processor mistaken as a faulty one. The pessimistic diagnosability of a system G, denoted by , is the maximal number of faulty processors so that the system G is t/t-diagnosable. The known results about for alternating group graphs [Inform. Process. Lett. (2015), pp. 151–154]; BC networks [IEEE Trans. Comput. (2005), pp. 176–184]; the k-ary n-cube networks [IEEE Trans. Comput. (1991), pp. 232–237], [Int. J. Comput. Math. (2012), pp. 1–10] etc. have property that . In this paper, we study the pessimistic diagnosability of two kinds of graphs with , those are: bubble-sort star graphs and augmented k-ary n-cubes , and prove that for , for , and for and .

Let F be a faulty set in an n-dimensional hypercube Q n such that in Q n - F each vertex is incident to at least two edges, and let f v , f e be the numbers of faulty vertices and faulty edges in F, respectively. In this paper, we consider the fault-tolerant edge-bipancyclicity of hypercubes. It is shown that each edge in Q n - F for n ? 3 lies on a fault-free cycle of any even length from 6 to 2 n - 2 f v if f v + f e ? 2 n - 5 . This gives an answer for a problem proposed by Yang et al. (2016) 33. Consider hypercubes with mixed faulty edges and faulty vertices under conditional-fault model.Improve the previous results about the fault-tolerant edge-bipancyclicity of hypercubes.Give an affirmative answer for a problem proposed by Yang et al. (2016).

GRIM-19 (gene associated with retinoid-interferon-induced mortality 19), a novel cell death regulatory gene, plays important roles in cell apoptosis, mitochondrial respiratory chain and immune response. It has been reported to interact physically with STAT3 and inhibit STAT3-dependent signal transduction. In this study, a new GRIM-19 gene, which is a 789-bp gene encoding a 149 amino acids protein, is identified and characterized from Litopenaeus vannamei. The tissue distribution patterns showed that LvGRIM-19 was widely expressed in all examined tissues, with the highest expression in muscle. Quantitative real-time PCR revealed that LvGRIM-19 was down-regulated in hepatopancreas after infection with the Vibrio alginolyticus. Knockdown of LvGRIM-19 by RNA interference resulted in a lower mortality of L. vannamei under V. alginolyticus infection, as well as an enhancement in the protein expression of STAT gene and JAK gene. V. alginolyticus infection caused an increase apoptotic cell ratio and ROS production of L. vannamei, while LvGRIM-19 silenced shrimps showed significantly lower than GFP group. Our results suggest that the GRIM-19 plays a vital role in shrimps' responses to V. alginolyticus. Interferenced LvGRIM-19 treatment during V. alginolyticus infection could increase 12 h survival rate, which might indicated that LvGRIM-19 is closely related to death of shrimps.

The pessimistic diagnosis strategy is a classic strategy based on the PMC model. A system is t / t -diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free node mistaken as a faulty one. The pessimistic diagnosability of a system G, denoted by t p ( G ) , is the maximal number of faulty processors so that the system G is t / t -diagnosable. In this paper, we study the pessimistic diagnosabilities of some general k-regular k-connected graphs G n . The main result t p ( G n ) = 2 k - 2 - g under some conditions is obtained, where g is the maximum number of common neighbors between any two adjacent vertices in G n . As applications of the main result, every pessimistic diagnosability of many famous networks including some known results, such as the alternating group networks AN n , the k-ary n-cubes Q n k , the star graphs S n , the matching composition networks G ( G 1 , G 2 ; M ) and the alternating group graphs AG n , are obtained. We study the pessimistic diagnosabilities of some general regular graphs under the PMC model.The exact value of t p ( G n ) is obtained.As applications, the pessimistic diagnosability of AN n , Q n k , S n , some MCN and AG n etc., are obtained.

Given a graph G and a non-negative integer g, the g-extraconnectivity of G is the cardinality of a minimum set of vertices in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with more than g vertices. The 2-extraconnectivity of k-ary n-cubes is gotten by Hsieh and Chang [Extraconnectivity of k-ary n-cube networks. Theoret. Comput. Sci. 443 2012 63–69] for . This paper shows that the 3-extraconnectivity of the k-ary n-cubes is , where and .

The data center networks $D_{n,k}$, proposed in 2008, has many desirable features such as high network capacity. A kind of generalization of diagnosability for network $G$ is $g$-good-neighbor diagnosability which is denoted by $t_g(G)$. Let $\kappa^g(G)$ be the $R^g$-connectivity. Lin et. al. in [IEEE Trans. on Reliability, 65 (3) (2016) 1248--1262] and Xu et. al in [Theor. Comput. Sci. 659 (2017) 53--63] gave the same problem independently that: the relationship between the $R^g$-connectivity $\kappa^g(G)$ and $t_g(G)$ of a general graph $G$ need to be studied in the future. In this paper, this open problem is solved for general regular graphs. We firstly establish the relationship of $\kappa^g(G)$ and $t_g(G)$, and obtain that $t_g(G)=\kappa^g(G)+g$ under some conditions. Secondly, we obtain the $g$-good-neighbor diagnosability of $D_{k,n}$ which are $t_g(D_{k,n})=(g+1)(k-1)+n+g$ for $1\leq g\leq n-1$ under the PMC model and the MM model, respectively. Further more, we show that $D_{k,n}$ is tightly super $(n+k-1)$-connected for $n\geq 2$ and $k\geq 2$ and we also prove that the largest connected component of the survival graph contains almost all of the remaining vertices in $D_{k,n}$ when $2k+n-2$ vertices removed., Comment: 16 pages, 2 figures

Extra connectivity and the pessimistic diagnosis are two crucial subjects for a multiprocessor system's ability to tolerate and diagnose faulty processor. The pessimistic diagnosis strategy is a classic strategy based on the PMC model in which isolates all faulty vertices within a set containing at most one fault-free vertex. In this paper, the result that the pessimistic diagnosability $t_p(G)$ equals the extra connectivity $\kappa_{1}(G)$ of a regular graph $G$ under some conditions are shown. Furthermore, the following new results are gotten: the pessimistic diagnosability $t_p(S_n^2)=4n-9$ for split-star networks $S_n^2$, $t_p(\Gamma_n)=2n-4$ for Cayley graphs generated by transposition trees $\Gamma_n$, $t_p(\Gamma_{n}(\Delta))=4n-11$ for Cayley graph generated by the $2$-tree $\Gamma_{n}(\Delta)$, $t_{p}(BP_n)=2n-2$ for the burnt pancake networks $BP_n$. As corollaries, the known results about the extra connectivity and the pessimistic diagnosability of many famous networks including the alternating group graphs, the alternating group networks, BC networks, the $k$-ary $n$-cube networks etc. are obtained directly., Comment: 23 pages

Let G be a connected graph and S be a set of vertices. The h-extra connectivity of G is the cardinality of a minimum set S such that G − S is disconnected and each component of G − S has at least h + 1 vertices. The h-extra connectivity for h = 1 , 2 of k-ary n-cubes are gotten by Hsieh and Chang (2012) [14] for k ≥ 4 and Zhu et al. (2011) [20] for k = 3 . In this paper, we show that the h-extra connectivity of the 3-ary n-cubes for h = 3 is equal to 8 n − 12 , where n ≥ 3 .

Prohibitin2 (PHB2), a potential tumor suppressor protein, plays important roles in inhibition of cell cycle progression, transcriptional regulation, apoptosis and the mitochondrial respiratory chain. To explore its potential roles in crustaceans' immune responses we have identified and characterized LvPHB2, a 891 bp gene encoding a 297 amino acids protein in the shrimp Litopenaeus vannamei. Expression analyses showed that LvPHB2 is expressed in all examined tissues, and largely present in cytoplasm, correlating with its known anti-oxidation function in mitochondria. Luciferase reporter assays showed that over-expression of LvPHB2 could activate the p53 pathway, indicating that it might participate in apoptosis regulation. Quantitative real-time PCR revealed that infection with Vibrio alginolyticus induces its up-regulation in hepatopancreas. Moreover, RNAi knock-down of LvPHB2 in vivo raises mortality rates of L. vannamei infected by V. alginolyticus, and affects expression of STAT3, Caspase3 and p53 genes. We found significantly higher reactive oxygen species production, DNA damage and apoptosis rates in LvPHB2-silenced shrimp challenged with V. alginolyticus than in controls injected with a Green Fluorescent Protein-silencing construct. Our results suggest that LvPHB2 plays a vital role in shrimp responses to V. alginolyticus infection through its participation in regulation of oxidants and apoptosis.

Cadmium, one of the most toxic heavy metals in aquatic environments, has severe effects on marine invertebrates and fishes. The MAPK signaling pathway plays a vital role in stress responses of animals. The mitogen-activated protein kinase (MAPK) signaling pathway plays a vital role in animals' stress responses, including mediation of apoptosis induced by the Rho GTPase Cdc42. However, there is limited knowledge about its function in shrimps, although disorders exacerbated by environmental stresses (including heavy metal pollution) have caused serious mortality in commercially cultured shrimps. Thus, we probed roles of Cdc42 in Litopenaeus vannamei shrimps (LvCdc42) during cadmium exposure by inhibiting its expression using dsRNA-mediated RNA interference. The treatment successfully reduced expression levels of MAPKs (including p38, JNK, and ERK). Cadmium exposure induced significant increases in expression levels of LvCdc42 and MAPKs, accompanied by reductions in total hemocyte counts (THC) and increases in apoptotic hemocyte ratios and ROS production. However, all of these responses were much weaker in LvCdc42-suppressed shrimps, in which mortality rates were higher than in controls. Our results suggest that the MAPK pathway plays a vital role in shrimps' responses to Cd(2+). They also indicate that LvCdc42 in shrimps participates in its regulation, and thus plays key roles in ROS production, regulation of apoptosis and associated stress responses.

The g-good-neighbor conditional diagnosability is the maximum number of faulty vertices a network can guarantee to identify, under the condition that every fault-free vertex has at least g fault-free neighbors. In this paper, we study the 1-good-neighbor conditional diagnosabilities of some general k-regular k-connected graphs G under the PMC model and the MM* model. The main result [Formula: see text] under some conditions is obtained, where l is the maximum number of common neighbors between any two adjacent vertices in G. Moreover, the following results are derived: [Formula: see text] for the hierarchical star networks, [Formula: see text] for the BC networks, [Formula: see text] for the alternating group graphs [Formula: see text].

A system is t/t-diagnosable if, provided the number of faulty processor is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free node mistake as a faulty one. The pessimistic diagnosability of a system G, denoted by tp(G), is the maximal number of faulty processors so that the system G is t/t-diagnosable. The augmented cube AQn, proposed by Choudum and Sunitha [Networks 40 (2) (2002) 71–84], has many attractive properties such as regularity, strong connectivity and symmetry. In this paper, we determine the pessimistic diagnosability of the n-dimensional augmented cube AQn and prove that tp(AQn)=4n−8 for n≥5.

The balanced hypercube BHn, proposed by Wu and Huang, is a new variation of hypercube. A Hamiltonian bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path between two arbitrary vertices from different partite sets. A Hamiltonian laceable graph G is strongly Hamiltonian laceable if there is a path of length [Formula: see text] between any two distinct vertices of the same partite set. A graph G is called k-edge-fault strong Hamiltonian laceable, if G – F is strong Hamiltonian laceable for any edge-fault set F with [Formula: see text]. It has been proved that the balanced hypercube BHn is strong Hamiltonian laceable. In this paper, we improve the above result and prove that BHn is (n – 1)-edge-fault strong Hamiltonian laceable.

Let tc(G) and tg(G) be the conditional diagnosability and g-good-neighbor diagnosability, respectively, of a graph G. The notion of the g-good-neighbor conditional diagnosability is less restrictive as compared with that of the conditional diagnosability in general. Particularly, the conditional faulty set notion requires that, any vertex, faulty or not, have at least one non-faulty neighbor; while the 1-good-neighbor faulty only requires that a non-faulty vertex have at least one non-faulty neighbor. Compared with conditional diagnosability, g-good-neighbor diagnosability is interesting since it characterizes a stronger tolerance capability. In this paper, we investigate the equal relation between t1(BHn) and tc(BHn) for the balanced hypercubes BHn. That is [Formula: see text] for [Formula: see text] under the PMC model and [Formula: see text] for [Formula: see text] under the MM model; Furthermore, the 2-good-neighbor diagnosability t2(BHn) = 4n − 1 for n ≥ 2 under the PMC model and the MM model is obtained.

Let Fv (resp. Fe) be the set of faulty vertices (resp. faulty edges) in the n-dimensional balanced hypercube BHn. The edge-bipancyclicity of BHn − Fv for |Fv| ≤ n − 1 had been proved in [Inform. Sci. 288 (2014) 449–461]. The existence of edge-Hamiltonian cycles in BHn − Fe for |Fe| ≤ 2n − 2 were obtained in [Appl. Math. Comput. 244 (2014) 447–456]. In this paper, we consider fault-tolerant cycle embedding of BHn with both faulty vertices and faulty edges, and prove that there exists a fault-free cycle of length 22n − 2|Fv| in BHn with |Fv| + |Fe| ≤ 2n − 3 and |Fv| ≤ n − 1 for n ≥ 2.