Searcharxiv⌕ Search

arXiv subjects

Martin Knor

Publications and source records attributed to Martin Knor.

25 records · Page 2Linked to original sources
arXiv2020

Graphs with the edge metric dimension smaller than the metric dimension

Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance vectors to such a set. In this work, we settle three open problems on (edge) metric dimension of graphs. Specifically, we show that for every $r,t\ge 2$ with $r\ne t$, there is $n_0$, such that for every $n\ge n_0$ there exists a graph $G$ of order $n$ with metric dimension $r$ and edge metric dimension $t$, which among other consequences, shows the existence of infinitely many graph whose edge metric dimension is strictly smaller than its metric dimension. In addition, we also prove that it is not possible to bound the edge metric dimension of a graph $G$ by some constant factor of the metric dimension of $G$.

math.CO↗
arXiv2019

The structure of graphs with given number of blocks and the maximum Wiener index

The Wiener index (the distance) of a connected graph is the sum of distances between all pairs of vertices. In this paper, we study the maximum possible value of this invariant among graphs on $n$ vertices with fixed number of blocks $p$. It is known that among graphs on $n$ vertices that have just one block, the $n$-cycle has the largest Wiener index. And the $n$-path, which has $n-1$ blocks, has the maximum Wiener index in the class of graphs on $n$ vertices. We show that among all graphs on $n$ vertices which have $p\ge 2$ blocks, the maximum Wiener index is attained by a graph composed of two cycles joined by a path (here we admit that one or both cycles can be replaced by a single edge, as in the case $p=n-1$ for example).

cs.DM↗
arXiv2019

Graphs with the second and third maximum Wiener index over the 2-vertex connected graphs

Wiener index, defined as the sum of distances between all unordered pairs of vertices, is one of the most popular molecular descriptors. It is well known that among 2-vertex connected graphs on $n\ge 3$ vertices, the cycle $C_n$ attains the maximum value of Wiener index. We show that the second maximum graph is obtained from $C_n$ by introducing a new edge that connects two vertices at distance two on the cycle if $n\ne 6$. If $n\ge 11$, the third maximum graph is obtained from a $4$-cycle by connecting opposite vertices by a path of length $n-3$. We completely describe also the situation for $n\le 10$.

cs.DM↗
arXiv2017

On the minimum value of sum-Balaban index

We consider extremal values of sum-Balaban index among graphs on $n$ vertices. We determine that the upper bound for the minimum value of the sum-Balaban index is at most $4.47934$ when $n$ goes to infinity. For small values of $n$ we determine the extremal graphs and we observe that they are similar to dumbbell graphs, in most cases having one extra edge added to the corresponding extreme for the usual Balaban index. We show that in the class of balanced dumbbell graphs, those with clique sizes $\sqrt[4]{\sqrt 2\log\big(1+\sqrt 2\big)}\sqrt n+o(\sqrt n)$ have asymptotically the smallest value of sum-Balaban index. We pose several conjectures and problems regarding this topic.

math.CO↗
arXiv2016

On the difference between the Szeged and Wiener index

We prove a conjecture of Nadjafi-Arani, Khodashenas and Ashrafi on the difference between the Szeged and Wiener index of a graph. Namely, if $G$ is a 2-connected non-complete graph on $n$ vertices, then $Sz(G)-W(G)\ge 2n-6$. Furthermore, the equality is obtained if and only if $G$ is the complete graph $K_{n-1}$ with an extra vertex attached to either $2$ or $n-2$ vertices of $K_{n-1}$. We apply our method to strengthen some known results on the difference between the Szeged and Wiener index of bipartite graphs, graphs of girth at least five, and the difference between the revised Szeged and Wiener index. We also propose a stronger version of the aforementioned conjecture.

math.CO↗
arXiv2015

Mathematical aspects of Wiener index

The Wiener index (i.e., the total distance or the transmission number), defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. In this article we summarize some results, conjectures and problems on this molecular descriptor, with emphasis on works we were involved in.

math.CO↗