We present a block diagonalization method for the adjacency matrices of two types of covering graphs. A graph \(Y\) is a covering graph of a base graph \(X\) if there exists an onto graph map \(\pi: Y \to X\) such that for each \(x \in X\) and for each \(y \in \{y \mid \pi(y) = x\}\), the collection of vertices adjacent to \(y\) maps onto the collection of vertices adjacent to \(x \in X\). The block diagonalization method requires the irreducible representations of the Galois group of \(Y\) over \(X\). The first type of covering graph is the Cayley graph over the finite ring \(\mathbb{Z}/p^n\mathbb{Z}\). The second type of covering graph resembles large lattices with vertices \(\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}\) for large \(n\). For one lattice, the block diagonalization method allows us to obtain explicit formulas for the eigenvalues of its adjacency matrix. We use these formulas to analyze the distribution of its eigenvalues. For another lattice, the block diagonalization method allows us to find non-trivial bounds on its eigenvalues.

Broadcast domination in graphs is a variation of domination in which different integer weights are allowed on vertices and a vertex with weight \(k\) dominates its distance \(k\)-neighborhood. A distribution of weights on vertices of a graph \(G\) is called a dominating broadcast, if every vertex is dominated by some vertex with positive weight. The broadcast domination number \(\gamma_b(G)\) of a graph \(G\) is the minimum weight (the sum of weights over all vertices) of a dominating broadcast of \(G\). In this paper, we prove that for a connected graph \(G\), \(\gamma_b(G) \geq \lceil{2\text{rad}(G)}/{3}\rceil\). This general bound and a newly introduced concept of condensed dominating broadcast are used in obtaining sharp upper bounds for broadcast domination numbers of three standard graph products in terms of broadcast domination numbers of factors. A lower bound for a broadcast domination number of the Cartesian product of graphs is also determined, and graphs that attain it are characterized. Finally, as an application of these results, we determine exact broadcast domination numbers of Hamming graphs and Cartesian products of cycles.

The semigirth \(\gamma\) of a digraph \(D\) is a parameter related to the number of shortest paths in \(D\). In particular, if \(G\) is a graph, the semigirth of the associated symmetric digraph \(G^*\) is \(\ell(G^*) = \lfloor {g(G) – 1}/{2} \rfloor\), where \(g(G)\) is the girth of the graph \(G\). In this paper, some bounds for the minimum number of vertices of a \(k\)-regular digraph \(D\) having girth \(g\) and semigirth \(\ell\), denoted by \(n(k, g; \ell)\), are obtained. Moreover, we construct a family of digraphs which achieve the lower bound for some particular values of the parameters.

For a graph \(G\), let \(\mathcal{D}(G)\) be the set of all strong orientations of \(G\). Define the orientation number of \(G\), \(\overrightarrow{d}(G) = \min\{d(D) \mid D \in \mathcal{D}(G)\}\), where \(d(D)\) denotes the diameter of the digraph \(D\). In this paper, it has been shown that \(\overrightarrow{d}(G \times H) = d(G)\), where \(\times\) denotes the tensor product of graphs, \(H\) is a special type of circulant graph, and the diameter, \(d(G)\), of \(G\) is at least \(4\). Some interesting results have been obtained using this result. Further, it is shown that \(d(P_r \times K_s) = d(P_r)\) for suitable \(r\) and \(s\). Moreover, it is proved that \(\overrightarrow{d}(C_r \times K_s) = d(C_r)\) for appropriate \(r\) and \(s\).

We consider some partitions where even parts appear twice and some where evens do not repeat. Further, we offer a new partition theoretic interpretation of two mock theta functions of order \(8\).

A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. The purpose of this paper is to generalize some known theorems and results of cordial graphs. Specifically, we show that certain combinations of paths, cycles, and stars are cordial.

An edge-magic total labeling on a graph with \(p\) vertices and \(q\) edges is defined as a one-to-one map taking the vertices and edges onto the integers \(1, 2, \ldots, p+q\) with the property that the sum of the labels on an edge and of its endpoints is constant, independent of the choice of edge. The magic strength of a graph \(G\), denoted by \(emt(G)\), is defined as the minimum of all constants over all edge-magic total labelings of \(G\). The maximum magic strength of a graph \(G\), denoted by \(eMt(G)\), is defined as the maximum constant over all edge-magic total labelings of \(G\). A graph \(G\) is called weak magic if \(eMt(G) – emt(G) > p\). In this paper, we study some classes of weak magic graphs.

In the first part of this paper, we present a generalization of complete graph factorizations obtained by labeling the graph vertices by natural numbers. In this generalization, the vertices are labeled by elements of an arbitrary group \(G\), in order to achieve a \(G\)-transitive factorization of the graph.

Vertex colorings of Steiner systems \(S(t,t+1,v)\) are considered in which each block contains at least two vertices of the same color. Necessary conditions for the existence of such colorings with given parameters are determined, and an upper bound of the order \(O(\ln v)\) is found for the maximum number of colors. This bound remains valid for nearly complete partial Steiner systems, too. In striking contrast, systems \(S(t,k,v)\) with \(k \geq t+2\) always admit colorings with at least \(c\cdot v^\alpha\) colors, for some positive constants \(c\) and \(\alpha\), as \(v\to\infty\).

Cwatsets were originally defined as subsets of \(\mathbb{Z}_2^d\) that are “closed with a twist.” Attempts have been made to generalize them, but the generalizations have failed to produce notions of subcwatset and quotient cwatset that behave naturally.

We present a new, abstract definition that appears to avoid these problems. The relationship between this new definition and its predecessor is similar to that between the abstract definition of “group” and its original meaning as a set of permutations. To justify the broader definition, we use small cancellation theory to prove a result analogous to the statement that every group is isomorphic to some permutation group. After developing the notion of a quotient cwatset, we prove an analogue of the First Homomorphism Theorem.