The Picard group is defined as \( \Gamma = SL(2, \mathbb{Z}[i]) \); the ring of \( 2 \times 2 \) matrices with Gaussian integer entries and determinant one. We consider certain graphs associated to quotients \( \Gamma/\Gamma(p) \) where \( p \) is a prime congruent to three mod four and \( \Gamma(p) \) is the congruence subgroup of level \( p \). We prove a decomposition theorem on the vertices of these graphs, and use this decomposition to derive upper and lower bounds on their isoperimetric numbers.

The domination number of a graph \( G \), \( \gamma(G) \), and the domination graph of a digraph \( D \), \( dom(D) \), are integrated in this paper. The \( \gamma \)-set di domination graph of the complete biorientation of a graph \( G \), \( dom_{\gamma}(\overset{\leftrightarrow}{G}) \), is created. All \( \gamma \)-sets of specific trees \( T \) are found, and \( dom_{\gamma}(\overset{\leftrightarrow}{T}) \) is characterized for those classes.

A fractional automorphism of a graph is a doubly stochastic matrix which commutes with the adjacency matrix of the graph. If we apply an ordinary automorphism to a set of vertices with a particular property, such as being independent or dominating, the resulting set retains that property. We examine the circumstances under which fractional automorphisms preserve the fractional properties of functions on the vertex set.

A king graph \( KG_n \) has \( n^2 \) vertices corresponding to the \( n^2 \) squares of an \( n \times n \) chessboard. From one square (vertex) there are edges to all squares (vertices) being attacked by a king. For given graphs \( G \) and \( H \), the Ramsey number \( r(G, H) \) is the smallest \( n \) such that any 2-coloring of the edges of \( KG_n \) contains \( G \) in the first or \( H \) in the second color. Results on existence and nonexistence of \( r(G, H) \) and some exact values are presented.

A set \( \{a_1,a_2,\ldots,a_n\} \) of positive integers with \( a_1 < a_2 < \cdots < a_n \) is said to be equi-graphical if there exists a graph with exactly \( a_i \) vertices of degree \( a_i \) for each \( i \) with \( 1 \leq i \leq n \). It is known that such a set is equi-graphical if and only if \( \sum_{i=1}^{n} a_i \) is even and \( a_n \leq \sum_{i=1}^{n-1} a_i^2 \). This concept is generalized to the following problem: Given a set \( S \) of positive integers and a permutation \( \pi \) on \( S \), determine when there exists a graph containing exactly \( a_i \) vertices of degree \( \pi(a_i) \) for each \( i \) (\( 1 \leq i \leq n \)). If such a graph exists, then \( \pi \) is called a graphical permutation. In this paper, the graphical permutations on sets of size four are characterized and using a criterion of Fulkerson, Hoffman, and McAndrew, we show that a permutation \( \pi \) of \( S = \{a_1,a_2,\ldots,a_n\} \), where \( 1 \leq a_1 < a_2 < \cdots < a_n \) and such that \( \pi(a_n) = a_n \), is graphical if and only if \( \sum_{i=1}^{n} a_i\pi(a_i) \) is even and \( a_n \leq \sum_{i=1}^{n-1} a_i\pi(a_i) \).

The formula for the number of spanning trees in \( K_{t_1,\ldots,t_P} \) is well known. In this paper, we give an algorithm that generates the list of spanning trees in \( K_{s,t} \).

For any \( k \in \mathbb{N} \), a graph \( G = (V,E) \) is said to be \( \mathbb{Z}_k \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_k – \{0\} \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_k \) defined by
\[
l^+(v) = \sum_{u \in N(v)} l(uv)
\]
is a constant map. For a given graph \( G \), the set of all \( k \in \mathbb{Z}_+ \) for which \( G \) is \( \mathbb{Z}_k \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). In this paper, we will consider trees whose diameters are at most \( 4 \) and will determine their integer-magic spectra.