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.

An eigenvalue of a graph G is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Let G 0 be the graph obtained from G by deleting all pendant vertices and δ(G) the minimum degree of vertices of G. In this paper, all connected tricyclic graphs G with δ(G 0) ≥ 2 and exactly two main eigenvalues are determined.

Let $G=\left(V\right(G),E(G\left)\right)$ be a graph. Gould and Hynds (1999) showed a well-known characterization of $G$ by its line graph $L\left(G\right)$ that has a 2-factor. In this paper, by defining two operations, we present a characterization for a graph $G$ to have a 2-factor in its line graph $L\left(G\right).$ A graph $G$ is called $N^2$-locally connected if for every vertex $x\in V\left(G\right),$$G[...$