A graph is called of type k if it is connected, regular, and has k distinct eigenvalues. For example graphs of type 2 are the complete graphs, while those of type 3 are the strongly regular graphs. We prove that for any positive integer n, every graph can be embedded in n cospectral, non-isomorphic graphs of type k for every k ≥ 3. Furthermore, in the case k ≥ 5 such a family of extensions can ...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using 5 colors. This result is tight since there are...

We establish the existence of vertex-magic total labelings (VMTLs) for several infinite classes of regular graphs. The main method of construction is to assemble a number of appropriately labeled copies of one graph into a single graph with a VMTL. This method enables us for example to begin with any even-regular graph and from it construct a cubic graph possessing a VMTL. An important feature ...

Recently the following question was relayed [1] to the second author: What is the cardinality of the set of vertex transitive graphs of finite degree? Our aim in this short note is to show that there are 20 such graphs. Our proof is constructive and is based on ideas of B. Neumann [3]. In order to construct a large such set it is natural to turn to Cayley graphs of finitely generated groups (se...