The Resolving Graph of Amalgamation of Cycles

For an ordered set W = {w_1,w_2,...,w_k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v,w_1),d(v,w_2),...,d(v,w_k)) where d(x,y) represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by dim(G), is the number of vertices in a basis of G. A resolving set W of G is connected if the subgraph induced by W is a nontrivial connected subgraph of G. The connected resolving number is the minimum cardinality of a connected resolving set in a graph G, denoted by cr(G). A cr-set of G is a connected resolving set with cardinality cr(G). A connected graph H is a resolving graph if there is a graph G with a cr-set W such that = H. Let {G_i} be a finite collection of graphs and each G_i has a fixed vertex v_{oi} called a terminal. The amalgamation Amal{Gi,v_{oi}} is formed by taking of all the G_i's and identifying their terminals. In this paper, we determine the connected resolving number and characterize the resolving graphs of amalgamation of cycles

On regular fuzzy resolving set

In a fuzzy graph G, if the degree of each vertex is the same, then it is called a regular fuzzy graph. The representation of σ − H with respect to the subset H of σ are all distinct then H is called the resolving set of the fuzzy graph G(V, σ, µ). In this article, we define a regular fuzzy resolving set, regular fuzzy resolving number and the properties of a regular fuzzy resolving set in a fuzzy graph whose crisp graph is a cycle, even or odd. And also we prove that, if G be a regular fuzzy graph with G* is a cycle, then any minimum fuzzy resolving set of G is a regular fuzzy resolving set of G.Emerging Sources Citation Index (ESCI)MathScinetScopu

Metric dimension of dual polar graphs

A resolving set for a graph $\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension $\mu(\Gamma)$ is the smallest size of a resolving set for $\Gamma$. We consider the metric dimension of the dual polar graphs, and show that it is at most the rank over $\mathbb{R}$ of the incidence matrix of the corresponding polar space. We then compute this rank to give an explicit upper bound on the metric dimension of dual polar graphs.Comment: 8 page