The Wiener number of a graph G is defined as 1/2∑d(u,v), where u,v ∈ V(G), and d is the distance function on G. The Wiener number has important applications in chemistry. We determine the Wiener number of an important family of graphs, namely, the Kneser graphs.

The Wiener number of a graph G is defined as $1/2{\sum }_{u,v\in V\left(G\right)}d(u,v)$, d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, $W\left(\mu \left(S{ₙ}^k\right)\right)\le W\left(\mu \left(T{ₙ}^k\right)\right)\le W\left(\mu \left(P{ₙ}^k\right)\right)$, where Sₙ, Tₙ and Pₙ denote a star, a general tree and a path on n vertices respectively. We also obtain Nordhaus-Gaddum type inequality for the Wiener number of $\mu \left(G^k\right)$.

In this paper, by a travel groupoid is meant an ordered pair $(V,*)$ such that $V$ is a nonempty set and $*$ is a binary operation on $V$ satisfying the following two conditions for all $u,v\in V$: $$(u*v)*u=u;\text{if}(u*v)*v=u,\text{then}u=v.$$ Let $(V,*)$ be a travel groupoid. It is easy to show that if $x,y\in V$, then $x*y=y$ if and only if $y*x=x$. We say that $(V,*)$ is on a (finite or infinite) graph $G$ if $V\left(G\right)=V$ and $$E\left(G\right)=\left\{\right\{u,v\}u,v\in V\text{and}u\ne u*v=v\}.$$ Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.

The notion of travel groupoids was introduced by L. Nebeský in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set $V$ and a binary operation $*$ on $V$ satisfying two axioms. We can associate a graph with a travel groupoid. We say that a graph $G$ has a travel groupoid if the graph associated with the travel groupoid is equal to $G$. Nebeský gave a characterization of finite graphs having a travel groupoid. In this paper, we study travel groupoids on infinite graphs....

Tree-like isometric subgraphs of hypercubes, or tree-like partial cubes as we shall call them, are a generalization of median graphs. Just as median graphs they capture numerous properties of trees, but may contain larger classes of graphs that may be easier to recognize than the class of median graphs. We investigate the structure of tree-like partial cubes, characterize them, and provide examples of similarities with trees and median graphs. For instance, we show that the cube graph of a tree-like...

Tree-like partial cubes were introduced in [B. Brešar, W. Imrich, S. Klavžar, Tree-like isometric subgraphs of hypercubes, Discuss. Math. Graph Theory, 23 (2003), 227-240] as a generalization of median graphs. We present some incorrectnesses from that article. In particular we point to a gap in the proof of the theorem about the dismantlability of the cube graph of a tree-like partial cube and give a new proof of that result, which holds also for a bigger class of graphs, so called tree-like partial...

We prove that every triangle-free planar graph with minimum degree 3 has radius at least 3; equivalently, no vertex neighborhood is a dominating set.

A graph $G$ is called an $S$-graph if its periphery $Peri\left(G\right)$ is equal to its center eccentric vertices $Cep\left(G\right)$. Further, a graph $G$ is called a $D$-graph if $Peri\left(G\right)\cap Cep\left(G\right)=\varnothing$. We describe $S$-graphs and $D$-graphs for small radius. Then, for a given graph $H$ and natural numbers $r\ge 2$, $n\ge 2$, we construct an $S$-graph of radius $r$ having $n$ central vertices and containing $H$ as an induced subgraph. We prove an analogous existence theorem for $D$-graphs, too. At the end, we give some properties of $S$-graphs and $D$-graphs.