A (1,≤ 2)-identifying code is a subset of the vertex set C of a graph such that each pair of vertices intersects C in a distinct way. This has useful applications in locating errors in multiprocessor networks and threat monitoring. At the time of writing, there is no simply-stated rule that will indicate if a graph is (1,≤ 2)-identifiable. As such, we discuss properties that must be satisfied b...

A dominating set S of a graph G is called locating-dominating, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number λ(G). An LD-set S of a graph G is global if it is an LD-set of both G and its complem...

Several recent studies of complex networks have suggested algorithms for locating network communities, also called modules or clusters, which are mostly defined as groups of nodes with dense internal connections. Along with the rapid development of these clustering techniques, the ability of revealing overlaps between communities has become very important as well. An efficient search technique ...

This article analyses Locating Adverbials (LAs) such as un peu plus tard, ce matin, deux kilomètres plus loin (‘a little later’, ‘this morning’, ‘two kilometers further’) when they are dislocated to the left of the sentence (IP Adjuncts cases). Although not discourse connectives, in such a position, they seem to play an important part in structuring discourse. It is this contribution of LAs to ...

solvable graphs (also known as reachable graphs) are types of graphs that any arrangement of a specified number of agents located on the graph’s vertices can be reached from any initial arrangement through agents’ moves along the graph’s edges, while avoiding deadlocks (interceptions). in this paper, the properties of solvable graphs are investigated, and a new concept in multi agent moti...

let $n$ be any positive integer and let $f_n$ be the friendship (or dutch windmill) graph with $2n+1$ vertices and $3n$ edges. here we study graphs with the same adjacency spectrum as the $f_n$. two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. let $g$ be a graph cospectral with $f_n$. here we prove that if $g$ has no cycle of length $4$ or $...

the first zagreb index $m_1$ of a graph $g$ is equal to the sum of squaresof degrees of the vertices of $g$. goubko proved that for trees with $n_1$pendent vertices, $m_1 geq 9,n_1-16$. we show how this result can beextended to hold for any connected graph with cyclomatic number $gamma geq 0$.in addition, graphs with $n$ vertices, $n_1$ pendent vertices, cyclomaticnumber $gamma$, and minimal $m...

the total version of geometric–arithmetic index of graphs is introduced based on the endvertexdegrees of edges of their total graphs. in this paper, beside of computing the total gaindex for some graphs, its some properties especially lower and upper bounds are obtained.

We introduce the notion of a centroidal locating set of a graph G, that is, a set L of vertices such that all vertices in G are uniquely determined by their relative distances to the vertices of L. A centroidal locating set of G of minimum size is called a centroidal basis, and its size is the centroidal dimension CD(G). This notion, which is related to previous concepts, gives a new way of ide...

A partition Π of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of Π. The partition dimension of G is the minimum cardinality of a locating partition of G. A pair of vertices u, v of a graph G are called twins if they have exactly the same set of neighbors other than u and v. A twin class is a ...