The vulnerability of the communication network measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. Cable cuts, node interruptions, software errors or hardware failures and transmission failure at various points can cause interrupt service for long periods of time. High levels of service dependability have traditionally characterised communication services. In communication networks, requiring greater degrees of stability or less vulnerability. If we think of graph G as modelling a network, the neighbour-integrity and edge-neighbour-integrity of a graph, which are considered as the neighbour vulnerability, are two measures of graph vulnerability. In the neighbour-integrity, it is considered that any failure vertex affects its neighbour vertices. In the edge-neighbour-integrity it is consider that any failure edge affects its neighbour edges.
In this paper we study classes of recursive graphs that are used to design communication networks and represent the molecular structure, and we show neighbour-integrity (vertex and edge) among the recursive graphs.
Let $\Gamma$ be a rectifiable Jordan curve in the finite complex plane $\mathbb C$ which is regular in the sense of Ahlfors and David. Denote by $L^2_C (\Gamma)$ the space of all complex-valued functions on $\Gamma$ which are square integrable w.r. to the arc-length on $\Gamma$. Let $L^2(\Gamma)$ stand for the space of all real-valued functions in $L^2_C (\Gamma)$ and put
\[ L^2_0 (\Gamma) = \lbrace h \in L^2 (\Gamma)\; \int _{\Gamma} h(\zeta ) |\mathrm{d}\zeta | =0\rbrace. \] Since the Cauchy singular operator is bounded on $L^2_C (\Gamma)$, the Neumann-Poincaré operator $C_1^{\Gamma}$ sending each $h \in L^2 (\Gamma)$ into \[ C_1^{\Gamma} h(\zeta _0) := \Re (\pi \mathrm{i})^{-1} \mathop {\mathrm P. V.}\int _{\Gamma} \frac{h(\zeta )}{\zeta -\zeta _0} \mathrm{d}\zeta , \quad \zeta _0 \in \Gamma , \] is bounded on $L^2(\Gamma)$. We show that the inclusion
\[ C_1^{\Gamma} (L^2_0 (\Gamma)) \subset L^2_0 (\Gamma)
\] characterizes the circle in the class of all $AD$-regular Jordan curves $\Gamma$.
We prove and discuss some new (Hp,Lp)-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients {q_{k}:k\geqslant 0}. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of general Nörlund means tn with non-increasing coefficients analogous results can be obtained for Fejér and Cesàro means by choosing the generating sequence {q_{k}:k\geqslant 0} in an appropriate way., István Blahota, Lars-Erik Persson, Giorgi Tephnadze., and Obsahuje seznam literatury
We study $n$-dimensional $QR$-submanifolds of $QR$-dimension $(p-1)$ immersed in a quaternionic space form $QP^{(n+p)/4}(c)$, $c\geqq 0$, and, in particular, determine such submanifolds with the induced normal almost contact $3$-structure.