Definition 1. Given a undirected graph G = (V,E), two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = (v1, v2, . . . , vn) ∈ V n such that vi is adjacent to vi+1 for 1 ≤ i < n. Such a path P is called a path of length n from v1 to vn. Let ei,j be the edge incident to both vi and vj. Given a real-valued weight functi...

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices of N there is no monochromatic path between them and for every vertex v / ∈ N there is a monochromatic path from v to N . We denote...

We prove that the C *-algebra of a directed graph E is liminal iff the graph satisfies the finiteness condition: if p is an infinite path or a path ending with a sink or an infinite emitter, and if v is any vertex, then there are only finitely many paths starting with v and ending with a vertex in p. Moreover, C * (E) is Type I precisely when the circuits of E are either terminal or transitory,...

We consider trees, series-parallel digraphs, and general series-parallel digraphs that have vertex weights and delays. The length/delay of a path is the sum of the delays on the path. We show that minimal weight vertex subsets X such that the length of the longest path is bounded by a given value δ when all vertices in X are upgraded to have delay 0 can be found in pseudo polynomial time. In ca...

Given an undirected graph G and vertices s, t ∈ V (G), the undirected connectivity problem is to decide whether there is a path from s to t in G. If G has N vertices and M edges, then the existence of a path between s and t implies the existence of a path of length at most N . If A is the adjacency matrix of G and G has, without loss of generality, self-loops on every vertex, then s and t are c...

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψk(G) the minimum cardinality of a k-path vertex cover in G. In this paper improved lower and upper bounds for ψk of the Cartesian and the direct product of paths are derived. It is shown that for ψ3 those bounds are tight. For the lexicographic produc...

We prove that the C *-algebra of a directed graph E is liminal iff the graph satisfies the finiteness condition: if p is an infinite path or a path ending with a sink or an infinite emitter, and if v is any vertex, then there are only finitely many paths starting with v and ending with a vertex in p. Moreover, C * (E) is Type I precisely when the circuits of E are either terminal or transitory,...

Directed s-t connectivity is the problem of detecting whether there is a path from vertex s to vertex t in a directed graph. We present the rst known deterministic sublinear space, polynomial time algorithm for directed s-t connectivity. For n-vertex graphs, our algorithm can use as little as n=2 ( p logn) space while still running in polynomial time.

In this study, suborbital graphs, $G_{u,N}$ and $F_{u,N}$ are examined. Modular group $\Gamma$ its act on $\widehat{\mathbb{Q}}$ studied. Lorentz matrix that gives the vertices obtained under classical multiplication in graph is analysed with multiplication. written as Möbius transform normalized type of researched. Moreover, a different element scrutinized. The path starting $\infty$ For path,...

An r-simple k-path is a path in the graph of length k that passes through each vertex at most r times. The r-SIMPLE k-PATH problem, given a graph G as input, asks whether there exists an r-simple k-path in G. We first show that this problem is NP-Complete. We then show that there is a graph G that contains an r-simple k-path and no simple path of length greater than 4 log k/ log r. So this, in ...