Related papers: The Median of Sierpinski Triangle Graphs

Given a simple connected undirected graph G, the Wiener index W(G) of G is defined as half the sum of the distances over all pairs of vertices of G. In practice, G corresponds to what is known as the molecular graph of an organic compound.…

The Wiener index $W(G)$ is the sum of distances of all pairs of vertices of the graph $G$. The Wiener polarity index $W_{p}(G)$ of a graph $G$ is the number of unordered pairs of vertices $u$ and $v$ of $G$ such that the distance…

A strict lower bound for the diameter of a symmetric graph is proposed, which is calculable with the order $n$ and other local parameters of the graph such as the degree $k\,(\geq 3)$, even girth $g\,(\geq 4)$, and number of $g$-cycles…

A simple-triangle graph (also known as a PI graph) is the intersection graph of a family of triangles defined by a point on a horizontal line and an interval on another horizontal line. The recognition problem for simple-triangle graphs was…

The flip-graph of a convex polygon $\pi$ is the graph whose vertices are the triangulations of $\pi$ and whose edges correspond to flips between them. The eccentricity of a triangulation $T$ of $\pi$ is the largest possible distance in this…

Using existing technology, we prove a Masur-Minsky style distance formula for flip- graph distance between two triangulations, expressed as a sum of the distances of the projections of these triangulations into arc graphs of the suitable…

The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the…

In several recent papers, the maximal safety distance that two players can maintain while moving through a graph has been defined and studied using three different spans of the graph, each with different movement conditions. Mainly, vertex…

Persistence diagrams are common objects in the field of Topological Data Analysis. They are topological summaries that capture both topological and geometric structure within data. Recently there has been a surge of interest in developing…

The Sierpi\'{n}ski product $G \otimes _f H$ of graphs $G$ and $H$ with respect to a function $f \colon V(G)\rightarrow V(H)$ has the vertex set $V(G)\times V(H)$. For every $g\in V(G)$ it contains a disjoint copy $gH$ of $H$, and for every…

The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an $n$-vertex maximal planar graph is at most $\lfloor\frac{1}{18}(n^3+3n^2)\rfloor$.…

A vertex k-ranking is a labeling of the vertices of a graph with integers from 1 to k so any path connecting two vertices with the same label will pass through a vertex with a greater label. The rank number of a graph is defined to be the…

The first Zagreb index $M_{1}$ of a graph is defined as the sum of the square of every vertex degree, and the second Zagreb index $M_{2}$ of a graph is defined as the sum of the product of vertex degrees of each pair of adjacent vertices.…

The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph. This is a generalization of eccentric connectivity…

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$. A triangular grid graph is a subgraph of a tiling of the plane with…

Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with…

A variable line through the centroid G of a triangle divides the triangle into two parts each of whose lengths as a fraction of the perimeter fills a closed interval [m,1-m], with m between 0 and 1/2. We show that the range of m taken over…

The outer multiset dimension ${\rm dim}_{\rm ms}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that…

Let $G=(V,E)$ be a connected graph. A vertex $w\in V$ distinguishes two elements (vertices or edges) $x,y\in E\cup V$ if $d_G(w,x)\ne d_G(w,y)$. A set $S$ of vertices in a connected graph $G$ is a mixed metric generator for $G$ if every two…

Given a Poisson process on a bounded interval, its random geometric graph is the graph whose vertices are the points of the Poisson process and edges exist between two points if and only if their distance is less than a fixed given…