In this paper, we characterize the potentially \((K_5 – C_4)\)-graphic sequences, where \(K_s – C_4\) is the graph obtained from \(K_5\) by removing four edges of a \(4\)-cycle \(C_4\). This characterization implies a theorem due to Lai \([6]\).

A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. The purpose of this paper is to generalize some known theorems and results of cordial graphs. Specifically, we show that certain combinations of paths, cycles, stars, and null graphs are cordial. Finally, we prove that the torus grids are cordial if and only if its size is not congruent to \(2\) \((mod 4)\).

A graph \(G\) is edge-magic if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1, 2, 3, \ldots, |V(G)| + |E(G)|\}\) such that for any edge \(uv\) of \(G\), \(f(u) + f(uv) + f(v)\) is constant. Moreover, \(G\) is super edge-magic if \(V(G)\) receives \(\{1, 2, \ldots, |V(G)|\}\) smallest labels. In this paper, we propose methods for constructing new (super) edge-magic graphs from some old ones by adding some new pendant edges.

In this study, we consider a generalization of the well-known Fibonacci and Lucas numbers related to combinatorial sums by using finite differences. To write generalized Fibonacci and Lucas sequences in a new direct way, we investigate some new properties of these numbers.

A graph \(G\) is called edge-magic if there exists a bijective function \(\phi: V(G) \cup E(G) \rightarrow \{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that \(\phi(x) + \phi(xy) + f\phi(y) = c(\phi)\) is a constant for every edge \(xy \in E(G)\), called the valence of \(\phi\). A graph \(G\) is said to be super edge-magic if \(\phi(V(G)) = \{1, 2, \ldots, |V(G)|\}\). The super edge-magic deficiency, denoted by \(\mu_s(G)\), is the minimum nonnegative integer \(n\) such that \(G \cup nK_1\) has a super edge-magic labeling, if such integer does not exist we define \(\mu_s(G)\) to be \(+\infty\). In this paper, we study the super edge-magic deficiency of some families of unicyclic graphs.

In \([FP]\) the \(ECO\) methed and Aigner’s theory of Catalan-like numbers are compared, showing that it is often possible to translate a combinatorial situation from one theory into the other by means of a standard change of basis in a suitable vector space. In the present work we emphasize the soundness of such an approach by finding some applications suggested by the above mentioned translation. More precisely, we describe a presumably new bijection between two classes of lattice paths and we give a combinatorial interpretation to an integer sequence not appearing in \([SI]\).

High stopping-distance low-density parity-check \((LDPC)\) product codes with finite geometry \(LDPC\) and Hamming codes as the constituent codes are constructed. These codes have high stopping distance compared to some well-known LDPC codes. As examples, linear \((511, 180, 30)\), \((945, 407, 27)\), \((2263, 1170, 30)\), and \((4095, 2101, 54)\) LDPC codes are designed with stopping distances \(30\), \(27\), \(30\), and \(54\), respectively. Due to their good stopping redundancy, they can be considered as low-complexity codes with very good performance when iterative decoding algorithms are used.

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.