We give decomposition formulas of the multiedge and the multipath zeta function of a regular covering of a graph \( G \) with respect to equivalence classes of prime, reduced cycles of \( G \). Furthermore, we give a decomposition formula of the weighted zeta function of a \( g \)-cyclic \( \Gamma \)-cover of a symmetric digraph \( D \) with respect to equivalence classes of prime cycles of \( D \), for any finite group \( \Gamma \) and \( g \in \Gamma \).

Let \( A \) be an abelian group. We call a graph \( G = (V, E) \) \( A \)-magic if there exists a labeling \( f : E(G) \to A^* \) such that the induced vertex set labeling \( f^+ : V(G) \to A \), defined by \( f^+(v) = \sum_{(u,v) \in E(G)} f(u,v) \), is a constant map. In this paper, we present some algebraic properties of \( A \)-magic graphs. Using them, various results are obtained for group-magic eulerian graphs.

Every Latin square of prime or prime power order \( s \) corresponds to a polynomial in 2 variables over the finite field on \( s \) elements, called the local permutation polynomial. What characterizes this polynomial is that its restrictions to one variable are permutations. We discuss the general form of local permutation polynomials and prove that their total degree is at most \( 2s – 4 \), and that this bound is sharp. We also show that the degree of the local permutation polynomial for Latin squares having a particular form is at most \( s – 2 \). This implies that circulant Latin squares of prime order \( p \) correspond to local permutation polynomials having degree at most \( p – 2 \). Finally, we discuss a special case of circulant Latin squares whose local permutation polynomial is linear in both variables.

Two graphs are said to be flow-equivalent if they have the same number of nowhere-zero \( \lambda \)-flows, i.e., they have the same flow polynomial. In this paper, we present a few methods of constructing non-isomorphic flow-equivalent graphs.

The Whitney number \( W_m{(n,k)} \) of the rank-\( n \) Dowling lattice \( Q_n(G) \) based on the group \( G \) having order \( m \) is the number of elements in \( Q_n(G) \) of co-rank \( k \). The associated numbers \( U_m{(n,k)} = k! W_m{(n,k)} \) and \( V_m{(n,k)} = k! m^k W_m{(n,k)} \) were studied by M. Benoumhani [\({Adv. in Appl. Math}\). 19 (1997), no. 1, 106-116] where a generating function was derived using algebraic techniques and logconcavity was shown for \( \{U_m{(n,k)}\} \) and for \( \{V_m{(n,k)}\} \). We give a central limit theorem and a local limit theorem on \( \mathbb{R} \) for \( \{U_m{(n,k)}\} \) and for \( \{V_m{(n,k)}\} \). In addition, asymptotic formulas for \( \max_k U_m{(n,k)} \), \( \max_k V_m{(n,k)} \) and their modes are given.

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 \( a_n \leq \sum_{i=1}^{n-1} a_i\pi(a_i) \).