We introduce a process where connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When vertex splits, incoming edges are assigned between offspring and Poisson random number of added them. The is parametrised positive real λ which governs the limiting average degree. show that for each value there unique M(λ) invariant under this evolut...

Titania nanotubes (TNT) prepared by anodization of Ti foils were used for water splitting in a standalone cell. The concentration polarization between the anode side (1M NaOH) and cathode side (0.5 M H2SO4) ensured that the water splitting reaction could take place with no external bias and separate H2 and O2 evolution could be achieved. The destructi...

many graph theoretical parameters have been used to describe the vulnerability of communication networks, including toughness, binding number, rate of disruption, neighbor-connectivity, integrity, mean integrity, edgeconnectivity vector, l-connectivity and tenacity. in this paper we discuss integrity and its properties in vulnerability calculation. the integrity of a graph g, i(g), is defined t...

For a graph G, a detachment operation at a vertex transforms the graph into a new graph by splitting the vertex into several vertices in such a way that the original graph can be obtained by contracting all the split vertices into a single vertex. A graph obtained from a given graph G by applying detachment operations at several vertices is called a detachment of graph G. We consider a detachme...

The graph scheme developed in 1 is further investigated. The structure of the coefficients aais determined in terms of a0 and the projections of the splitting graphs. This allows a formal sum­ mation of the /c-series. The first few terms of the strong coupling series for the lowest mass eigen­ value are computed for the X <}r4 theory. The leading term for large /. is complex. This restricts phy...

We introduce the notion of uniform number of a graph. The  uniform number of a connected graph $G$ is the least cardinality of a nonempty subset $M$ of the vertex set of $G$ for which the function $f_M: M^crightarrow mathcal{P}(X) - {emptyset}$ defined as $f_M(x) = {D(x, y): y in M}$ is a constant function, where $D(x, y)$ is the detour distance between $x$ and $y$ in $G$ and $mathcal{P}(X)$ ...

If M is a compact Riemannian manifold then we show that for typical continuous function defined on M, the upper box dimension of  graph(f) is as big as possible and the lower box dimension of graph(f) is as small as possible.  

Many graph theoretical parameters have been used to describe the vulnerability of communication networks, including toughness, binding number, rate of disruption, neighbor-connectivity, integrity, mean integrity, edgeconnectivity vector, l-connectivity and tenacity. In this paper we discuss Integrity and its properties in vulnerability calculation. The integrity of a graph G, I(G), is defined t...