In this paper, we obtain the numbers of embeddings of wheel graphs on some orientable and nonorientable surfaces of small genera, mainly on torus, double torus, and nonorientable surfaces of genus \(1, 2, 3\), and \(4\). These are the first results for embeddings of wheel graphs on nonorientable surfaces as known up to now.

An \((a, d)\)-edge-antimagic total labeling for a graph \(G(V, E)\) is an injective mapping \(f\) from \(V \cup E\) onto the set \(\{1, 2, \ldots, |V| + |E|\}\) such that the set \(\{f(v) + \sum f(uv) \mid uv \in E\}\), where \(v\) ranges over all of \(V\), is \(\{a, a+d, a+2d, \ldots, a+(|V|-1)d\}\). Simanjuntak et al conjecture:1. \(C_{2n}\) has a \((2n + 3, 4)\)- or a \((2n + 4, 4)\)-edge-antimagic total labeling;
2. cycles have no \((a, d)\)-edge-antimagic total labelings with \(d > 5\).In this paper, these conjectures are shown to be true.

This article discusses the geometricity of the direct sum, direct product and lexicographic products of two lattices, and compute their characteristic polynomials and classify their geometricity.

This paper introduces the concepts of a \({supergraph}\) and \({graphical\; complexity}\) of a permutation group, intended as a tool for investigating the structure of concrete permutation groups. Basic results are established and some research problems suggested.

We given a two parameter generalization of identities of Carlitzand Gould involving products of binomial coefficients. The generalization involves Jacobi polynomials.

Consider a connected undirected graph \(G = (V, E)\) and an integer \(r \geq 1\). For any vertex \(v \in V\), let \(B_r(v)\) denote the ball of radius \(r\) centered at \(v\), i.e., the set of all vertices linked to \(v\) by a path of at most \(r\) edges. If for all vertices \(v \in V\), the sets \(B_r(v)\) are different, then we say that \(G\) is \(r\)-twin-free.

Studies have been made, e.g., on the number of edges or the minimum degree in one-twin-free graphs. We extend these investigations and in particular we determine the exact size of the largest clique in a connected \(r\)-twin-free graph.

Let \(D\) be a strongly connected digraph with order at least two. Let \(M(D)\) denote the middle digraph of \(D\), and let \(\kappa(D)\) and \(\lambda(D)\) denote the connectivity and arc-connectivity of \(D\), respectively. In this paper, we study super-arc-connected and super-connected middle digraphs and the spectrum of middle digraphs.

For a connected graph \(G\) of order \(p \geq 2\), a set \(S \subseteq V(G)\) is an \(x\)-geodominating set of \(G\) if each vertex \(v \in V(G)\) lies on an \(x\)-geodesic for some element \(y \in S\). The minimum cardinality of an \(x\)-geodominating set of \(G\) is defined as the \(\alpha\)-geodomination number of \(G\), denoted by \(g_x(G)\) or simply \(g_x(G)\). An \(x\)-geodominating set of cardinality \(g_x(G)\) is called a \(g_x(G)\)-set. A connected graph of order \(p\) with vertex geodomination numbers either \(p – 1\) or \(p – 2\) for every vertex is characterized. It is shown that there is no graph of order \(p\) with vertex geodomination number \(p – 2\) for every vertex. Also, for an even number \(p\) and an odd number \(n\) with \(1 \leq n \leq p – 1\), there exists a connected graph \(G\) of order \(p\) and \(g_x(G) = n\) for every vertex \(x \in G\), and for an odd number \(p\) and an even number \(n\) with \(1 \leq n \leq p – 1\), there exists a connected graph \(G\) of order \(p\) and \(g_x(G) = n\) for every vertex \(x \in G\). It is shown that for any integer \(n > 2\), there exists a connected regular as well as a non-regular graph \(G\) with \(g_x(G) = n\) for every vertex \(x \in G\). For positive integers \(r, d\) and \(n \geq 2\) with \(r \leq d \leq 2r\), there exists a connected graph \(G\) of radius \(r\), diameter \(d\) and \(g_x(G) = n\) for every vertex \(x \in G\). Also, for integers \(p, d\) and \(n\) with \(3 \leq d \leq p – 1, 1 \leq n \leq p – 1\) and \(p – d – n + 1 \geq 0\), there exists a graph \(G\) of order \(p\), diameter \(d\) and \(g_x(G) = n\) for some vertex \(x \in G\).

A graph is called \emph{biclaw-free} if it has no biclaw as an induced subgraph. Lai and Yao [Discrete Math., \(307 (2007) 1217\)] conjectured that every \(2\)-connected biclaw-free graph \(G\) with \(\delta(G) \geq 4\) has a spanning eulerian subgraph \(H\) with maximum degree \(\Delta(H) \leq 4\). In this note, the conjecture is answered in the negative.

Let \(G\) be a graph of order \(n\) and size \(m\). A \(\gamma\)-labeling of \(G\) is a one-to-one function \(f: V(G) \to \{0, 1, 2, \ldots, m\}\) that induces a labeling \(f’: E(G) \to \{1, 2, \ldots, m\}\) of the edges of \(G\) defined by \(f'(e) = |f(u) – f(v)|\) for each edge \(e = uv\) of \(G\). The value of a \(\gamma\)-labeling \(f\) is defined as

\[val(f) = \sum\limits_{e \in E(G)} f'(e).\]

The \(\gamma\)-spectrum of a graph \(G\) is defined as

\[spec(G) = \{val(f): f \text{ is a \(\gamma\)-labeling of } G\}.\]

The \(\gamma\)-spectra of paths, cycles, and complete graphs are determined.