A proper total coloring of a graph \(G\) is called Smarandachely adjacent vertex total coloring of graph if for any two adjacent and distinct vertices \(u\) and \(v\) in \(G\), the set of colors assigned to the vertices and the edges incident to \(u\) doesn’t contain the set of colors assigned to the vertices and the edges incident to \(v\), vice versa. The minimal number of colors required for a Smarandachely adjacent vertex total coloring of graph is called the Smarandachely adjacent vertex total chromatic number of graph. In this paper, we define a kind of \(3\)-regular Multilayer Cycle \(Re(n,m)\) and obtain the Smarandachely adjacent vertex total chromatic number of it.

A perfectly one-factorable (PIF) regular graph \(G\) is a graph admitting a partition of the edge-set into one-factors such that the union of any two of them is a Hamiltonian cycle. We consider the case in which \(G\) is a cubic graph. The existence of a PIF cubic graph is guaranteed for each admissible value of the number of vertices. We give conditions for determining PIF graphs within a subfamily of generalized Petersen graphs.

In this paper, we give the generalization \(\{G_{k,n}\}_{n\in N }\) of \(k\)-Fibonacci and \(k\)-Lucas numbers. After that, by using this generalization, some new algebraic properties on these numbers have been obtained.

Let \(K_q(n, R)\) denote the least cardinality of a \(q\)-ary code of length \(n\), such that every \(q\)-ary word of length \(n\) differs from at least one word in the code in at most \(R\) places. We use a method of Blass and Litsyn to derive the bounds \(K_4(5,2) \geq 14\) and \(K_4(6,2) \geq 32\).

Let \(d_{q}(n,k)\) be the maximum possible minimum Hamming distance of a linear \([n, k]\) code over \(\mathbb{F}_q\). Tables of best known linear codes exist for all fields up to \(q = 9\). In this paper, linear codes over \(\mathbb{F}_{11}\) are constructed for \(k\) up to \(7\). The codes constructed are from the class of quasi-twisted codes. These results show that there exists a \((78,8)\) arc in \(\text{PG}(2,11)\). In addition, the minimum distances of the extended quadratic residue codes of lengths \(76\), \(88\) and \(108\) are determined.

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.