In this paper, we study the metric dimension problem in maximal outerplanar graphs. Concretely, if \(\beta (G)\) denotes of a graph G order n, prove that \(2\le \beta (G) \le \lceil \frac{2n}{5}\rceil \) and bounds are tight. We also provide linear algorithms to decide whether is 2 build resolving set S size \(\lceil for G. Moreover, characterize all graphs with 2.

Let G be a graph with vertex set V(G), and let d(x,y) denote the length of shortest path between nodes x y in G. For positive integer k for distinct x,y∈V(G), dk(x,y)=min{d(x,y),k+1} Rk{x,y}={z∈V(G):dk(x,z)≠dk(y,z)}. A subset S⊆V(G) is k-truncated resolving if |S∩Rk{x,y}|≥1 any pair x,y∈V(G). The metric dimension, dimk(G), minimum cardinality over all sets G, usual dimension recovered when k+1 ...

Multiplicities corresponding to irreducible characters are defined for reality-based algebras. These algebras with a distinguished basis include fusion rings, C-algebras, and the adjacency algebras of finite association schemes. The definition of multiplicity generalizes that for schemes. For a broad class of these structures, which includes the adjacency algebras, it is proved that if all the ...

A family of connected graphs G is said to be a family with constant metric dimension if its metric dimension is finite and does not depend upon the choice of G in G. In this paper, we study the metric dimension of the generalized Petersen graphs P (n, m) for n = 2m + 1 and m ≥ 1 and give partial answer of the question raised in [9]: Is P (n, m) for n ≥ 7 and 3 ≤ m ≤ bn−1 2 c, a family of graphs...

Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations Ax = 0 for the 0-1 adjacency matrix A. A graph G is singular of nullity η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A is η(G). Necessary and sufficient conditions are determined for a graph to be singular in terms of admissible induced subgra...

In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bipartite graph from its adjacency matrix. Finally, we note that if we add a loop at every probe vertex of a probe interval graph, then the Ferrers dimension of the corresponding symmetric bipa...

Modifying the Hausdorff-Buseman metric, we obtain a compatible metric with the Fell-Matheron topology on the space of closed subsets of a locally compact Hausdorff second countable space. We also give an alternative expression of this metric, and two more compatible metrics, a metric of summation form and a modified Rockafellar-Wets metric. Through the study of the relation between the original...