Let \(P_n\) be a path with \(n\) vertices. \(P_n^k\), the \(k\)-th power of the path \(P_n\), is a graph on the same vertex set as \(P_n\), and the edges that join all vertices \(x\) and \(y\) if and only if the distance between them is at most \(k\). In this paper, the crossing numbers of \(P_n^k\) are studied. Drawings of \(P_n^k\) are presented and proved to be optimal for the case \(n \leq 8\) and for the case \(k \leq 4\).

A graph is said to be locally grid if the structure around each of its vertices is a \(3 x 3\) grid. As a follow up of the research initiated in \([8]\) and \([9]\) we prove that most locally grid graphs are uniquely determined by their Tutte polynomial.

Let \(P(G, \lambda)\) be the chromatic polynomial of a graph \(G\). A graph \(G\) is chromatically unique if for any graph \(H\), \(P(H, \lambda) = P(G, \lambda)\) implies \(H\) is isomorphic to \(G\). In his Ph.D. thesis, Zhao [Theorems 5.4.2 and 5.4.3] proved that for any positive integer \(t \geq 3\), the complete \(t\)-partite graphs \(K(p – k, p, p, \ldots, p)\) with \(p \geq k+2 \geq 4\) and \(K(p-k, p – 1, p, \ldots, p)\) with \(p \geq 2k \geq 4\) are chromatically unique. In this paper, by expanding the technique employed by Zhao, we prove that the complete \(t\)-partite graph \(K(p-k,\underbrace{ p -1, \ldots, p-1}, \underbrace{p, \ldots, p})\) is chromatically unique for integers \(p \geq k+2 \geq 4\) and \(t \geq d+3 \geq 3\).

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.