On \(Q(a)P(b)\)-Super Edge-graceful Graphs

Let \( a, b \) be two positive integers. For the graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \) with \( p = |V(G)| \) and \( q = |E(G)| \), we define two sets \( Q(a) \) and \( P(b) \) as follows:

\[
Q(a) =
\begin{cases}
\{\pm a, \pm(a+1), \ldots, \pm(a+\frac{q-2}{2})\} & \text{if } q \text{ is even} \\
\{0\} \cup \{\pm a, \pm(a+1), \ldots, \pm(a + (q-3)/{2})\} & \text{if } q \text{ is odd}
\end{cases}
\]

\[
P(b) =
\begin{cases}
\{\pm b, \pm(b+1), \ldots, \pm(b + (p-2)/{2})\} & \text{if } p \text{ is even} \\
\{0\} \cup \{\pm b, \pm(b+1), \ldots, \pm(b + (\frac{p-3}{2})/2)\} & \text{if } p \text{ is odd}
\end{cases}
\]

For the graph \( G \) with \( p = |V(G)| \) and \( q = |E(G)| \), \( G \) is said to be \( Q(a)P(b) \)-super edge-graceful (in short \( Q(a)P(b) \)-SEG), if there exists a function pair \( (f, f^+) \) which assigns integer labels to the vertices and edges; that is, \( f^+ : V(G) \to P(b) \), and \( f: E(G) \to Q(a) \) such that \( f^+ \) is onto \( P(b) \) and \( f \) is onto \( Q(a) \), and

\[
f^+(u) = \sum\{f(u,v) : (u,v) \in E(G)\}.
\]

We investigate \( Q(a)P(b) \) super edge-graceful graphs.