A locating-total dominating set (LTDS) S of a graph G is a total dominating set S of G such that for every two vertices u and v in V(G) − S, N(u)∩S ≠ N(v)∩S. The locating-total domination number ( ) l t G  is the minimum cardinality of a LTDS of G. A LTDS of cardinality ( ) l t G  we call a ( ) l t G  -set. In this paper, we determine the locating-total domination number for the special clas...

A locating-dominating set D of a graph G is dominating where each vertex not in has unique neighborhood D, and the Locating-Dominating Set problem asks if contains such bounded size. This known to be NP-hard even on restricted classes, as interval graphs, split planar bipartite subcubic graphs. On other hand, it solvable polynomial time for some trees and, more generally, graphs cliquewidth. Wh...

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...

iv Acknowledgements v List of Tables vii List of Graphs vii

the energy of a graph is equal to the sum of the absolute values of its eigenvalues. two graphs of the same order are said to be equienergetic if their energies are equal. we point out the following two open problems for equienergetic graphs. (1) although it is known that there are numerous pairs of equienergetic, non-cospectral trees, it is not known how to systematically construct any such pa...

A locating-dominating set a of graph G is a dominating set D of G with the additional property that every two distinct vertices outside D have distinct neighbors in D; that is, for distinct vertices u and v outside D, N(u) ∩D 6= N(v) ∩D where N(u) denotes the open neighborhood of u. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-dom...

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If u and v are vertices of a graph, then d(u, v) denotes the distance from u to v. Let S = {v1, v2, . . . , vk} be a set of vertices in a connected graph G. For each v ∈ V (G), the k-vector cS(v) is defined by cS(v) = (d(v, v1), d(v, v2), · · · , d(v, vk)). A dominating set S = {v1, v2, . . . , vk} in a connected graph G is a metric-locatingdominating set, or an MLD-set, if the k-vectors cS(v) ...