A complete arc of size \(q^2 – 1\) is constructed in the Moulton plane of order \(q^2\) for \(q \geq 5\) odd.

On the basis of the joint tree model initiated and comprehensively described by Liu, we obtain the genus distributions of double pear ladder graphs (a type of new \(3\)-regular graphs) in orientable surfaces.

The silicates are the largest, the most interesting and the most complicated class of minerals by far. The basic chemical unit of silicates is the \((\text{SiO}_4)\) tetrahedron. A silicate sheet is a ring of tetrahedrons which are linked by shared oxygen nodes to other rings in a two-dimensional plane that produces a sheet-like structure. We consider the silicate sheet as a fixed interconnection parallel architecture and call it a silicate network. We solve the Minimum Metric Dimension problem, which is NP-complete for general graphs.

A pebbling step on a graph consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. We consider all weight functions defined on the vertices of a graph that satisfy some property \({P}\). The \({P}\)-pebbling number of a graph is the minimum number of pebbles needed in an arbitrary initial configuration so that, for any such weight function, there is a sequence of pebbling moves at the end of which each vertex has at least as many pebbles as required by the weight function. Some natural properties on graph products are induced by properties defined on the factor graphs. In this paper, we give a bound for the \({P}’\)-pebbling number associated with a particular kind of product property \({P}’\) in terms of the \({P}_i\)-pebbling numbers associated with the factor properties \({P}_1\) and \({P}_2\). We do this by introducing color pebbling, which may be of interest in its own right.

The \(k\)-th isoperimetric edge connectivity \(\gamma_k(G) = \min\{|[U,\overline{U}]| : U \subset V(G), |U| \geq k\}\). A graph \(G\) with \(\gamma_k(G) = \beta_k(G)\) is said to be \(\gamma_k\)-optimal, where \(\beta_k(G) = \min\{|[U,\overline{U}]| : U \subset V(G), |U| = k\}\). Let \(G\) be a connected \(d\)-regular graph. Write \(L(G)\) and \(P_2(G)\) the line graph and the 2-path graph of \(G\), respectively. In this paper, we derive some sufficient conditions for \(L(G)\) and \(P_2(G)\) to be \(\gamma_k\)-optimal.

In 1968, Vizing conjectured that for any edge chromatic critical graph \(G = (V,E)\) with maximum degree \(\Delta\) and independence number \(\alpha(G)\), \(\alpha(G) \leq \frac{|V|}{2}\). This conjecture is still open. In this paper, we prove that \(\alpha(G) \leq \frac{3\Delta-2}{5\Delta-2}|V|\) for \(\Delta = 11, 12\) and \(\alpha(G) \leq \frac{11\Delta-30}{17\Delta-30}|V|\) for \(13 \leq \Delta \leq 29\). This improves the known bounds for \(\Delta \in \{11, 12, \ldots, 29\}\).

Consider a communication network \(G\) in which a limited number of edge (arc) and/or vertex faults \(F\) might occur. A routing \(\rho\), i.e. a fixed path between each pair of vertices, for the network must be chosen without knowing which components might become faulty. The diameter of the surviving route graph \(R(G, \rho)/F\), where \(R(G, \rho)/F\) is a digraph with the same vertices as \(G – F\) and a vertex \(x\) being adjacent to another vertex \(y\) if and only if \(\rho(x, y)\) avoids \(F\), could be an important measurement for the routing \(\rho\). In this paper, the authors consider the Cartesian product digraphs whose factors satisfy some given conditions and show that the diameter of the surviving route graph is bounded by three for any minimal routing \(\rho\) when the number of faults is less than some integer. This result is also useful for the Cartesian product graphs and generalizes some known results.

The Tribonacci Zeta functions are defined by \(\zeta_T(s) = \sum_{k=1}^{\infty} {T_{k}^{-s}}\). We discuss the partial infinite sum \(\sum_{n=1}^{\infty} {T_{k}^{-s}}\) for some positive integer \(n\). We also consider the continued fraction expansion including Tribonacci numbers.