Related papers: The Median of Sierpinski Triangle Graphs

We define special Hamiltonian-paths and special permutations of the up-facing dark tiles on a checked triangular grid related to the generalized Sierpi\'{n}ski Gasket. Our definitions and observations make possible the generalization of the…

Let $G=(V,E)$ be a connected graph and let $d(u,v)$ denote the distance between vertices $u,v \in V$. A metric basis for $G$ is a set $B\subseteq V$ of minimum cardinality such that no two vertices of $G$ have the same distances to all…

The computation of the distance between any two points of the Sierpinski Gasket with respect to the intrinsic metric has already been investigated by several authors. In the literature there is not an explicit formula using the code space…

Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n…

Computing a graph prototype may constitute a core element for clustering or classification tasks. However, its computation is an NP-Hard problem, even for simple classes of graphs. In this paper, we propose an efficient approach based on…

We define the average $k$-TSP distance $\mu_{tsp,k}$ of a graph $G$ as the average length of a shortest walk visiting $k$ vertices, i.e. the expected length of the solution for a random TSP instance with $k$ uniformly random chosen…

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $c$ such that the vertex set $V(G)$ can be partitioned into sets $X_1, . . . , X_c$, with the condition that vertices in $X_i$ have pairwise distance…

Let D be an acyclic orientation of the graph G. An arc of D is dependent if its reversal creates a directed cycle. Let m(G) denote the minimum number of dependent arcs over all acyclic orientations of G. For any k > 0, a generalized…

A box in Euclidean $k$-space is the Cartesian product $I_1\times I_2\times \cdots \times I_k$, where $I_j$ is a closed interval on the real line. The boxicity of a graph $G$, denoted by $\text{box}(G)$, is the minimum nonnegative integer…

The unit ball random geometric graph $G=G^d_p(\lambda,n)$ has as its vertices $n$ points distributed independently and uniformly in the $d$-dimensional unit ball, with two vertices adjacent if and only if their $l_p$-distance is at most…

We obtain the numbers of spanning trees on the Sierpinski gasket $SG_d(n)$ with dimension $d$ equal to two, three and four. The general expression for the number of spanning trees on $SG_d(n)$ with arbitrary $d$ is conjectured. The numbers…

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…

A matchstick graph is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph $(m;n)$-regular if every vertex has only degree $m$ or $n$. In…

A geometric graph is a graph whose vertices are points in general position in the plane and its edges are straight line segments joining these points. In this paper we give an $O(n^2 \log n)$ algorithm to compute the number of pairs of…

We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straight-line edges and assign each edge to a layer so that no two edges on the same layer cross. The…

The $g$-girth-thickness $\theta(g,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least $g$ whose union is $G$. In this paper, we determine the $6$-girth-thickness $\theta(6,K_n)$ of the complete graph $K_n$ in…

Recently, settling a question of Erd\H{o}s, Balogh and Pet\v{r}\'{i}\v{c}kov\'{a} showed that there are at most $2^{n^2/8+o(n^2)}$ $n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here we characterize the…

We bound the mean distance in a connected graph which is not a tree in function of its order $n$ and its girth $g$. On one hand, we show that mean distance is at most $\frac{n+1}{3}-\frac{g(g^2-4)}{12n(n-1)}$ if $g$ is even and at most…

The medial axis of a smoothly embedded surface in $\mathbb{R}^3$ consists of all points for which the Euclidean distance function on the surface has at least two minima. We generalize this notion to the mid-sphere axis, which consists of…

{\small The Wiener index $W(G)$ of a graph $G$ is the sum of the distances between all pairs of vertices in the graph. The Szeged index $Sz(G)$ of a graph $G$ is defined as $Sz(G)=\sum_{e=uv \in E}n_u(e)n_v(e)$ where $n_u(e)$ and $n_v(e)$…