The average solution of a TSP instance in a graph

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 vertices. We prove relations with the average $k$-Steiner distance and characterize the cases where equality occurs. We also give sharp bounds for $\mu_{tsp,k}(G)$ given the order of the graph.Comment: 9 pages, 3 figure

Maximum Wiener Indices of Unicyclic Graphs of Given Matching Number

In this article, we determine the maximum Wiener indices of unicyclic graphs with given number of vertices and matching number. We also characterize the extremal graphs. This solves an open problem of Du and Zhou.Comment: 14 pages, 9 figure

Corrigendum on Wiener index, Zagreb Indices and Harary index of Eulerian graphs

In the original article ``Wiener index of Eulerian graphs'' [Discrete Applied Mathematics Volume 162, 10 January 2014, Pages 247-250], the authors state that the Wiener index (total distance) of an Eulerian graph is maximized by the cycle. We explain that the initial proof contains a flaw and note that it is a corollary of a result by Plesn\'ik, since an Eulerian graph is $2$-edge-connected. The same incorrect proof is used in two referencing papers, ``Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs'' [Bull. Malays. Math. Sci. Soc. (2019) 42:67-78] and ``Harary index of Eulerian graphs'' [J. Math. Chem., 59(5):1378-1394, 2021]. We give proofs of the main results of those papers and the $2$-edge-connected analogues.Comment: 5 Pages, 1 Figure Corrigendum of 3 papers, whose titles are combine

Five results on maximizing topological indices in graphs

In this paper, we prove a collection of results on graphical indices. We determine the extremal graphs attaining the maximal generalized Wiener index (e.g. the hyper-Wiener index) among all graphs with given matching number or independence number. This generalizes some work of Dankelmann, as well as some work of Chung. We also show alternative proofs for two recents results on maximizing the Wiener index and external Wiener index by deriving it from earlier results. We end with proving two conjectures. We prove that the maximum for the difference of the Wiener index and the eccentricity is attained by the path if the order $n$ is at least $9$ and that the maximum weighted Szeged index of graphs of given order is attained by the balanced complete bipartite graphs.Comment: 13 pages, 4 figure

The minimum number of maximal independent sets in twin-free graphs

The problem of determining the maximum number of maximal independent sets in certain graph classes dates back to a paper of Miller and Muller and a question of Erd\H{o}s and Moser from the 1960s. The minimum was always considered to be less interesting due to simple examples such as stars. In this paper we show that the problem becomes interesting when restricted to twin-free graphs, where no two vertices have the same open neighbourhood. We consider the question for arbitrary graphs, bipartite graphs and trees. The minimum number of maximal independent sets turns out to be logarithmic in the number of vertices for arbitrary graphs, linear for bipartite graphs and exponential for trees. In the latter case, the minimum and the extremal graphs have been determined earlier by Taletski\u{\i} and Malyshev, but we present a shorter proof.Comment: 17 pages, 7 figures, 5 table