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.

An \((a, d)\)-edge-antimagic total labeling on a \((p, q)\)-graph \(G\) is a one-to-one map \(f\) from \(V(G) \cup E(G)\) onto the integers \(1, 2, \ldots, p+q\) with the property that the edge-weights, \(w(uv) = f(u) + f(v) + f(uv)\) where \(uv \in E(G)\), form an arithmetic progression starting from \(a\) and having common difference \(d\). Such a labeling is called \emph{super} if the smallest possible labels appear on the vertices. In this paper, we investigate the existence of super \((a, d)\)-edge-antimagic total labeling of the disjoint union of multiple copies of the complete tripartite graph and the disjoint union of stars.

Given a configuration of pebbles on the vertices of a graph \(G\), a pebbling move consists of taking two pebbles off a vertex \(v\) and putting one of them back on a vertex adjacent to \(v\). A graph is called \({pebbleable}\) if for each vertex \(v\) there is a sequence of pebbling moves that would place at least one pebble on \(v\). The \({pebbling\;number}\) of a graph \(G\), is the smallest integer \(m\) such that \(G\) is pebbleable for every configuration of \(m\) pebbles on \(G\). A graph \(G\) is said to be class \(0\) if the pebbling number of \(G\) is equal to the number of vertices in \(G\). We prove that \(Bi-wheels\), a class of diameter three graphs, are class \(0\).