The Szeged polynomial of a connected graph \(G\) is defined as \(S_z(G,x) = \sum_{e \in E(G)} x^{n_{u(e) n_v(e)}} \), where \(n_u(e)\) is the number of vertices of \(G\) lying closer to \(u\) than to \(v\), and \(n_v(e)\) is the number of vertices of \(G\) lying closer to \(v\) than to \(u\). Ashrafi et al. (On Szeged polynomial of a graph, Bull. Iran. Math. Soc. \(33 (2007) 37-46)\) proved that if \(|V(G)|\) is even, then \(\deg(S_z(G,x)) \leq \frac{1}{4}{|V(G)^{2}} |\). In this paper, we investigate the structure of graphs with an even number of vertices for which equality holds, and also examine equality for the sum of graphs.

In this paper we investigate the number of rooted loopless unicursal planar maps and present some formulae for such maps with up to three parameters: the number of edges and the valencies of the two odd vertices.

In this paper, we investigate the metric dimension of generalized Petersen graphs \(P(n,3)\), providing a partial answer to an open problem posed in [8]: whether \(P(n,m)\) for \(n \geq 7\) and \(3 \leq m \leq \left\lfloor \frac{n-1}{2} \right\rfloor\) constitutes a family of graphs with constant metric dimension. Specifically, we prove that the metric dimension of \(P(n,3)\) equals \(3\) for \(n \equiv 1 \pmod{6}\), \(n \geq 25\), and equals \(4\) for \(n \equiv 0 \pmod{6}\), \(n \geq 24\). For remaining cases, four judiciously chosen vertices suffice to resolve all vertices of \(P(n,3)\), implying \(\dim(P(n,3)) \leq 4\), except when \(n \equiv 2 \pmod{6}\), in which case \(\dim(P(n,3)) \leq 5\).

Using subspaces of the finite field \(GF(q^{2^k})\) over \(GF(q)\), we construct new classes of external difference families.

Let \(M = \{v_1, v_2, \ldots, v_n\}\) be an ordered set of vertices in a graph \(G\). Then, \((d(u, v_1), d(u, v_2), \ldots, d(u, v_n))\) is called the \(M\)-coordinates of a vertex \(u\) of \(G\). The set \(M\) is called a \({metric\; basis}\) if the vertices of \(G\) have distinct \(M\)-coordinates. A minimum metric basis is a set \(M\) with minimum cardinality. The cardinality of a minimum metric basis of \(G\) is called the minimum metric dimension. This concept has wide applications in motion planning and robotics. In this paper, we solve the minimum metric dimension problem for Illiac networks.

For a graph \(G\) and a non-zero real number \(\alpha\), the graph invariant \(S_\alpha(G)\) is the sum of the \(\alpha^th\) power of the non-zero signless Laplacian eigenvalues of \(G\). In this paper, we obtain sharp bounds of \(S_\alpha(G)\) for a connected bipartite graph \(G\) on \(n\) vertices and a connected graph \(G\) on \(n\) vertices having a connectivity less than or equal to \(k\), respectively, and propose some open problems for future research.

In this paper we determine the scores of locally transitive tournaments and conversely, for such score we construct all locally transitive tournments having this score. This allows us to establish, for a given matrix, a test for the locally transitive property.

A graph is called a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product \(G \times H\) of graphs \(G\) and \(H\) has vertex set \(V(G) \times V(H)\) and edge set \(E(G \times H) = \{ (g_i, h_s)(g_j, h_t) \mid g_ig_j \in E(G) \text{ and } h_sh_t \in E(H) \}\). We prove that the direct product \(M_m(G) \times M_n(H)\) of the generalized Mycielskians of \(G\) and \(H\) is a cover graph if and only if \(G\) or \(H\) is bipartite.

For a primitive digraph \(D\) of order \(n\) and a positive integer \(m\) such that \(1 \leq m \leq n\), we define the \(m\)-competition index of \(D\), denoted by \(k_m(D)\), as the smallest positive integer \(k\) such that distinct vertices \(v_1, v_2, \ldots, v_m\) exist for each pair of vertices \(x\) and \(y\) with \(x \rightarrow^k v_i\) and \(y \rightarrow^k v_i\) for \(1 \leq i \leq m\) in \(D\). In this paper, we investigate the \(m\)-competition index of regular or almost regular tournaments.

A digraph \(D\) with \(e\) edges is labeled by assigning a distinct integer value \(\theta(v)\) from \(\{0, 1, \ldots, e\}\) to each vertex \(v\). The vertex values, in turn, induce a value \(\theta(u,v) = \theta(v) – \theta(u) \mod (e + 1)\) on each edge \((u,v)\). If the edge values are all distinct and nonzero, then the labeling is called a \emph{graceful labeling} of a digraph. Bloom and Hsu conjectured in 1985 that “all unicyclic wheels are graceful.” In this paper, we prove this conjecture.