Search

The edge-fault-tolerance of networks is of great significance to the design and maintenance of networks. For any pair of vertices $u$ and $v$ of the connected graph $G$, if they are connected by $\min \{ \deg_G(u),\deg_G(v)\}$ edge-disjoint paths, then $G$ is strong Menger edge connected (SM-$\lambda$ for short). The conditional edge-fault-tolerance about the SM-$ \lambda$ property of $G$, written $sm_\lambda^r(G)$, is the maximum value of $m$ such that $G-F$ is still SM-$\lambda$ for any edge subset $F$ with $|F|\leq m$ and $\delta(G-F)\geq r$, where $\delta(G-F)$ is the minimum degree of $G-F$. Previously, most of the exact value for $sm_\lambda^r(G)$ is aimed at some well-known networks when $r\leq 2$, and a few of the lower bounds on some well-known networks for $r\geq 3$. In this paper, we firstly determine the exact value of $sm_\lambda^r(G)$ on class of hypercube-like networks (HL-networks for short, including hypercubes, twisted cubes, crossed cubes etc.) for a general $r$, that is, $sm_\lambda^r(G_n)=2^r(n-r)-n$ for every $G_n\in HL_n$, where $n\geq 3$ and $1\leq r \leq n-2$., Comment: 9 pages, 3 figures

As a generalization of the traditional connectivity, the g-component edge connectivity c{\lambda}g(G) of a non-complete graph G is the minimum number of edges to be deleted from the graph G such that the resulting graph has at least g components. Hypercube-like networks (HL-networks for short) are obtained by manipulating some pairs of edges in hypercubes, which contain several famous interconnection networks such as twisted cubes, Mobius cubes, crossed cubes, locally twisted cubes. In this paper, we determine the (g + 1)-component edge connectivity of the n-dimensional HL-networks., Comment: 8 pages and 4 figures

Hypercube is one of the most important networks to interconnect processors in multiprocessor computer systems. Different kinds of connectivities are important parameters to measure the fault tolerability of networks. Lin et al.\cite{LinStructure} introduced the concept of $H$-structure connectivity $\kappa(Q_n;H)$ (resp. $H$-substructure connectivity $\kappa^s(Q_n;H)$) as the minimum cardinality of $F=\{H_1,\dots,H_m\}$ such that $H_i (i=1,\dots,m)$ is isomorphic to $H$ (resp. $F=\{H'_1,\dots,H'_m\}$ such that $H'_i (i=1,\dots,m)$ is isomorphic to connected subgraphs of $H$) such that $Q_n-V(F)$ is disconnected or trivial. In this paper, we discuss $\kappa(Q_n;H)$ and $\kappa^s(Q_n;H)$ for hypercubes $Q_n$ with $n\geq 3$ and $H\in \{P_k,C_k|3\leq k\leq 2^{n-1}\}$. As a by-product, we solve the problem mentioned in \cite{ManeStructure}., Comment: 15 pages, 9 figures

Dariush Kiani et al.\cite{kiani2015unitary} claim to have found the unitary Cayley graph $Cay(M_{n}(F),GL_{n}(F))$ of matrix algebras over finite field $F$ is strongly regular only when $n=2$, but they have only considered two special cases, namely when $n = 2$ and $3$ and have failed to cover the general cases. In this paper, we prove that the unitary Cayley graph of matrix algebras over finite field $F$ is strongly regular if and only if $n=2$.

In this paper,we construct some directed strongly regular Cayley graphs on dihedral groups,these generalizes some earlier constructions.We also characterize some certain directed strongly regular Cayley graphs on dihedral groups $D_{p^\alpha}$,where $p$ is a prime and $\alpha\geqslant 1$ is a positive integer., Comment: 20 pages

The concept of directed strongly regular graphs (DSRG) was introduced by Duval in 1988 \cite{A}.In the present paper,we use representation theory of finite groups in order to investigate the directed strongly regular Cayley graphs.We first show that a Cayley graph $\mathcal{C}(G,S)$ is not a directed strongly regular graph if $S$ is a union of some conjugate classes of $G$.This generalizes an earlier result of Leif K.J{\o}rgensen \cite{J1} on abelian groups.Secondly,by using induced representations,we have a look at the Cayley graph $\mathcal{C}(N\rtimes_\theta H, N_1\times H_1)$ with $N_1\subseteq N$ and $H_1\subseteq H$,determining its characteristic polynomial and its minimal polynomial.Based on this result,we generalize the semidirect product method of Art M. Duval and Dmitri Iourinski in \cite{D} and obtain a larger family of directed strongly regular graphs.Finally,we construct some directed strongly regular Cayley graphs on dihedral groups,which partially generalize the earlier results of Mikhail Klin,Akihiro Munemasa,Mikhail Muzychuk,and Paul Hermann Zieschang in \cite{K1}.By using character theory,we also give the characterization of directed strongly regular Cayley graphs $\mathcal{C}(D_n,X\cup Xa)$ with $X\cap X^{(-1)}=\emptyset$., Comment: 27 pages

The concept of directed strongly regular graphs (DSRG) was introduced by Duval in "A Directed Graph Version of Strongly Regular Graphs" [Journal of Combinatorial Theory, Series A 47(1988)71-100]. Duval also provided several construction methods for directed strongly regular graphs. In this paper, We construct several new classes of directed strongly regular graphs which are obtained by using Kronecker matrix product, Semidirect product and Cayley coset graph. At the same time, using group representation, for two special cases, we give some other sufficient and necessary conditions of Cayley graphs to be DSRG. At last, we finish this paper with a discussion of some propositions of in(out)-neighbours and automorphism group in directed strongly regular graphs., Comment: 35 pages,2 figures