New parameters for studying graceful properties of graphs

The consecutively super edge-magic deficiency of graphs and related concepts

“…The following corollary gives us an inequality relating the strong alpha-number of a graph and its consecutively super edge-magic deficiency. This result is easily obtained by substituting c = 1 into the equation c = x + n + p − q − s + 2 given in the proof of Theorem 2.1 (see [27]).…”

“…With the above definitions in hand, we can now state the following result established in [27], which shows the connection between the alpha-number of a graph and its consecutively super edge-magic deficiency.…”

“…We now provide the definitions for a new type of parameter (called the alpha-number) introduced in [27] and its restriction. They play an important role in the study of consecutively super edge-magic deficiency.…”

“…We now provide the definition for a parameter introduced in [27], the consecutively super edgemagic deficiency, used to measure the closeness of a graph to be consecutively super edge-magic. The consecutively super edge-magic deficiency µ c (G) of a graph G is defined to be either the smallest nonnegative integer n with the property that G ∪ nK 1 is consecutively super edge-magic or +∞ if there exists no such integer n.…”

On the strong beta-number of galaxies with three and four components

“…There are other kinds of parameters that measure how close a graph is to being graceful. For further knowledge on the (strong) beta-number of graphs and related concepts, the authors suggest that the reader consults the results found in [6][7][8][9][10]. For the most recent advances on the mentioned conjectures, the authors also direct the reader to the papers [5,11].…”