The basis number of a graph \(G\) is defined to be the least positive integer \(d\) such that \(G\) has a \(d\)-fold basis for the cycle space of \(G\).

In this paper, we prove that the basis number of the Cartesian product of different ladders is exactly \(4\). However, if we apply Theorem \(4.1\) of Ali and Marougi \([4]\), which is stated in the introduction as Theorem \(1.1\), we find that the basis number of the circular and Möbius ladders with circular ladders and Möbius ladders is less than or equal to \(5\), and the basis number of ladders with circular ladders and circular ladders with circular ladders is at most \(4\).

It is shown that there are \(\binom{2n-r-1}{n-r}\) noncrossing partitions of an \(n\)-set together with a distinguished block of size \(r\), and \(\binom{n}{k-1}\binom{n-r-1}{k-2}\) of these have \(k\) blocks, generalizing a result of Béna on partitions with one crossing. Furthermore, specializing natural \(q\)-analogues of these formulae with \(q\) equal to certain \(d^{th}\) roots of unity gives the number of such objects having \(d\)-fold rotational symmetry.

In this paper, we introduce the concept of geodesic graph at a vertex of a connected graph and investigate its properties. We determine the bounds for the number of edges of the geodesic graph. We prove that an edge of a graph is a cut edge if and only if it is a cut edge of each of its geodesic graphs. Also, we characterize a bipartite graph as well as a geodetic graph in terms of its geodesic graph.

In this paper, we study the circular choosability recently introduced by Mohar \([5]\) and Zhu \([11]\). In this paper, we show that the circular choosability of planar graphs with girth at least \(\frac{10n+8}{3}\) is at most \(2 + \frac{2}{n}\), which improves the earlier results.

An orientation of a simple graph \(G\) is called an oriented graph. If \(D\) is an oriented graph, \(\delta(D)\) its minimum degree and \(\lambda(D)\) its edge-connectivity, then \(\lambda(D) \leq \delta(D)\). The oriented graph is called maximally edge-connected if \(\lambda(D) = \delta(D)\) and super-edge-connected, if every minimum edge-cut is trivial. If \(D\) is an oriented graph with the property that the underlying graph \(G(D)\) contains no complete subgraph of order \(p+1\), then we say that the clique number \(\omega(D)\) of \(D\) is less or equal \(p\).

In this paper, we present degree sequence conditions for maximally edge-connected and super-edge-connected oriented graphs \(D\) with clique number \(\omega(D) \leq p\) for an integer \(p \geq 2\).

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.