We study the independence number of the Cartesian product of binary trees and more general bipartite graphs. We give necessary and sufficient conditions on bipartite graphs under which certain upper and lower bounds on the independence number of the product are equal. A basic tool will be an algorithm for finding the independence number of a binary tree.

Multireceiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify the authenticity of the received message. In this paper, we construct two multireceiver authentication codes from symplectic geometry over finite fields. The parameters and the probabilities of deceptions of the codes are also computed.

We give determinant expressions of the zeta function and an \(L\)-function of a semiregular weighted bipartite graph. As an application, we present a decomposition formula for the weighted complexity of a semiregular weighted bipartite graph.

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.