On the edge metric dimension and Wiener index of the blow up of graphs

Let $G=(V,E)$ be a connected graph. The distance between an edge $e=xy$ and a vertex $v$ is defined as \T{d}(e,v)=\T{min}\{\T{d}(x,v),\T{d}(y,v)\}. A nonempty set $S \subseteq V(G)$ is an edge metric generator for $G$ if for any two distinct edges $e_1,e_2 \in E(G)$, there exists a vertex $s \in S$ such that \T{d}(e_1,s) \neq \T{d}(e_2,s). An edge metric generating set with the smallest number of elements is called an edge metric basis of $G$, and the number of elements in an edge metric basis is called the edge metric dimension of $G$ and it is denoted by \T{edim}(G). In this paper, we study the edge metric dimension of a blow up of a graph $G$, and also we study the edge metric dimension of the zero divisor graph of the ring of integers modulo $n$. Moreover, the Wiener index and the hyper-Wiener index of the blow up of certain graphs are computed

Sufficient conditions for certain structural properties of graphs based on Wiener-type indices

Let $G=(V,E)$ be a simple connected graph with the vertex set $V$and the edge set $E$. The Wiener-type invariants of $G=(V,E)$ can beexpressed in terms of the quantities W_{f}=\sum_{\{u,v\}\subseteqV}f(d_{G}(u,v)) for various choices of the function $f$, where$d_{G}(u,v)$ is the distance between vertices $u$ and $v$ in $G$. Inthis paper, we establish sufficient conditions based on Wiener-typeindices under which every path of length $r$ is contained in aHamiltonian cycle, and under which a bipartite graph on $n+m$(m>n) vertices contains a cycle of size $2n$

Discrete Geometry (hybrid meeting)

A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group