General degree distance of graphs

We generalize several topological indices and introduce the general degree distance of a connected graph $G$. For $a, b \in \mathbb{R}$, the general degree distance $DD_{a,b} (G) = \sum_{ v \in V(G)} [deg_{G}(v)]^a S^b_{G} (v)$, where $V(G)$ is the vertex set of $G$, $deg_G (v)$ is the degree of a vertex $v$, $S^b_{G} (v) = \sum_{ w \in V(G) \setminus \{ v \} } [d_{G} (v,w) ]^{b}$ and $d_{G} (v,w)$ is the distance between $v$ and $w$ in $G$. We present some sharp bounds on the general degree distance for multipartite graphs and trees of given order, graphs of given order and chromatic number, graphs of given order and vertex connectivity, and graphs of given order and number of pendant vertices

Abelian Cayley graphs of given degree and diameter 2 and 3

Please read abstract in the article.http://link.springer.comjournal/3732015-11-30hj201

The degree-diameter problem for claw-free graphs and hypergraphs

Please read abstract in the article.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-01182015-08-30hb201

Wiener index of trees of given order and diameter at most 6

The long-standing open problem of nding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order n and diameter at most 6.The National Research Foundation and the University of KwaZulu-Natal.http://journals.cambridge.org/action/displayJournal?jid=BAZ2015-06-30hb201

On the metric dimension of circulant graphs with 44 generators

Circulant graphs are Cayley graphs of cyclic groups and the metric dimension of circulant graphs with at most $3$ generators has been extensively studied especially in the last decade. We extend known results in the area by presenting the lower and the upper bounds on the metric dimension of circulant graphs with $4$ generators

General degree distance of graphs

We generalize several topological indices and introduce the general degree distance of a connected graph $G$. For $a, b \in \mathbb{R}$, the general degree distance $DD_{a,b} (G) = \sum_{ v \in V(G)} [deg_{G}(v)]^a S^b_{G} (v)$, where $V(G)$ is the vertex set of $G$, $deg_G (v)$ is the degree of a vertex $v$, $S^b_{G} (v) = \sum_{ w \in V(G) \setminus \{ v \} } [d_{G} (v,w) ]^{b}$ and $d_{G} (v,w)$ is the distance between $v$ and $w$ in $G$. We present some sharp bounds on the general degree distance for multipartite graphs and trees of given order, graphs of given order and chromatic number, graphs of given order and vertex connectivity, and graphs of given order and number of pendant vertices

Wiener index of trees of given order and diameter at most 6

The long-standing open problem of finding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order $n$ and diameter at most $6$. DOI: 10.1017/S000497271300081