‎‎let r be a non-commutative ring with unity‎. ‎the commuting graph‎ ‎of $r$ denoted by $gamma(r)$‎, ‎is a graph with a vertex set‎ ‎$rsetminus z(r)$ and two vertices $a$ and $b$ are adjacent if and only if‎ ‎$ab=ba$‎. ‎in this paper‎, ‎we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n in lbrace 4,5 rbrace$‎. it is shown that‎, ‎$gamma(r)$ is the disjoint ...

A bound for the (n,m)-mixed chromatic number in terms of the chromatic number of the square of the underlying undirected graph is given. A similar bound holds when the chromatic number of the square is replaced by the injective chromatic number. When restricted to n = 1 andm = 0 (i.e., oriented graphs) this gives a new bound for the oriented chromatic number. In this case, a slightly improved b...

The aggregation technique, dedicated to two-terminal series-parallel graphs (or TTSP-graphs) and introduced lately to solve the minimum piecewise linear cost tension problem, is adapted here to solve the minimum binary cost tension problem (or BCT problem). Even on TTSP-graphs, the BCT problem has been proved to be NP-complete. As far as we know, the aggregation is the only algorithm, with mixe...

Graphs of order n with fault-tolerant metric dimension have recently been characterized.This paper points out an error in the proof this characterization. We show that complete multipartite graphs also n, which provides infinite family counterexamples to Furthermore, we find exact values metric, edge mixed-metric dimensions, domination number, locating-dominating and metric-locating-dominating ...

The mixed metric dimension $$\textrm{mdim}(G)$$ of a graph G is the cardinality smallest set vertices that (metrically) resolves each pair elements from $$V(G)\cup E(G)$$ . We say max-mdim if $$\textrm{mdim}(G) = n(G)$$ It proved with $$n(G)\ge 7$$ contains vertex degree at least 5. Using strong product graphs and amalgamations, large families are constructed. one universal determined. cut boun...

Let G be a graph and let S(G), M(G), T(G) the subdivision, middle, total of G, respectively. dim(G), edim(G), mdim(G) metric dimension, edge mixed dimension In this paper, for subdivision it is proved that max{dim(G), edim(G)} ≤ mdim(S(G)) mdim(G). A family graphs Gn constructed which mdim(Gn) − mdim(S(Gn)) ≥ 2 holds shows inequality can strict, while cactus = For middle dim(M(G)) holds, if tre...

In most studies, street networks are considered as undirected graphs while one-way streets and their effect on shortest paths usually ignored. Here, we first study the empirical of in about 140 cities world. Their presence induces a detour that persists over wide range distances is characterized by nonuniversal exponent. The one-ways pattern then twofold: they mitigate local traffic certain are...

Fuzzy mixed graph (FMG) can be used to model some graphical intercommunication network systems if there are many directed and undirected relations between vertices. In inverse fuzzy graph, the membership value of edges greater than or equal minimum corresponding (IFMG), exist vertices it analyze problems real life such that values pair this article, concept IFMG is introduced first with its pro...