It is well known that 3–regular graphs with arbitrarily large girth exist. Three constructions are given that use the former to produce non-Hamiltonian 3–regular graphs without reducing the girth, thereby proving that such graphs with arbitrarily large girth also exist. The resulting graphs can be 1–, 2– or 3–edge-connected depending on the construction chosen. From the constructions arise (nai...

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components. We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called ∆-spaces are counterexamples to Brouwer’s Conjecture. Using J.I. Hall’s characteriz...

In this paper, we are studying vertex-magic total labelings (VMTLs) of simple graphs. By now much is known about methods for constructing VMTLs for regular graphs. Here we are studying non-regular graphs. We show how to construct labelings for several families of non-regular graphs, including graphs formed as the disjoint union of two other graphs already possessing VMTLs. We focus on condition...

A graph Γ with diameter d is strongly distance-regular if Γ is distanceregular and its distance-d graph Γd is strongly regular. The known examples are all the connected strongly regular graphs (i.e. d = 2), all the antipodal distanceregular graphs, and some distance-regular graphs with diameter d = 3. The main result in this paper is a characterization of these graphs (among regular graphs with...