Let \(F(x,y) = ax^2 + bxy + cy^2\) be a binary quadratic form of discriminant \(\Delta = b^2 – 4ac\) for \(a,b,c \in \mathbb{Z}\), let \(p\) be a prime number and let \(\mathbb{F}_p\) be a finite field. In this paper we formulate the number of integer solutions of cubic congruence \(x^3 + ax^2 + bx + c \equiv 0 \pmod{p}\) over \(\mathbb{F}_p\), for two specific binary quadratic forms \(F_1^k(x,y) = x^2 + kxy + ky^2\) and \(F_2^k(x,y) = kx^2 + kxy + k^2y^2\) for integer \(k\) such that \(1 \leq k \leq 9\). Later we consider representation of primes by \(F_1^k\) and \(F_2^k\).

A subset \(S \subseteq V(G)\) is independent if no two vertices of \(S\) are adjacent in \(G\). In this paper we study the number of independent sets which meets the set of leaves in a tree. In particular we determine the smallest number and the largest number of these sets among \(n\)-vertex trees. In each case we characterize the extremal graphs.

A graph \(G\) is called super edge-magic if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1,2,\ldots,|V(G)| + |E(G)|\}\) such that \(f(u) + f(v) + f(uv) = k\) is a constant for any \(uv \in E(G)\) and \(f(V(G)) = \{1,2,\ldots,|V(G)|\}\). Yasuhiro Fukuchi proved that the generalized Petersen graph \(P(n, 2)\) is super edge-magic for odd \(n \geq 3\). In this paper, we show that the generalized Petersen graph \(P(n,3)\) is super edge-magic for odd \(n \geq 5\).

For any integer \(k\), two tournaments \(T\) and \(T’\), on the same finite set \(V\) are \(k\)-similar, whenever they have the same score vector, and for every tournament \(H\) of size \(k\) the number of subtournaments of \(T\) (resp. \(T’\)) isomorphic to \(H\) is the same. We study the \(4\)-similarity. According to the decomposability, we construct three infinite classes of pairs of non-isomorphic \(4\)-similar tournaments.

In this paper, we define the Pell and Pell-Lucas \(p\)-numbers and derive the analytical formulas for these numbers. These formulas are similar to Binet’s formula for the classical Pell numbers.

A graph \(G\) is called resonant if the boundary of each face of \(G\) is an \(F\)-alternating closed trail with respect to some \(f\)-factor \(F\) of \(G\). We show that a plane bipartite graph \(G\) is resonant if and only if it is connected and each edge of \(G\) is contained in an \(f\)-factor and not in another \(f\)-factor.

Let \(P_k\) denote a path with \(k\) vertices and \(k-1\) edges. And let \(\lambda K_{n,n}\) denote the \(\lambda\)-fold complete bipartite graph with both parts of size \(n\). A \(P_k\)-decomposition \(\mathcal{D}\) of \(\lambda K_{n,n}\) is a family of subgraphs of \(\lambda K_{n,n}\) whose edge sets form a partition of the edge set of \(\lambda K_{n,n}\), such that each member of \(\mathcal{G}\) is isomorphic to \(P_k\). Necessary conditions for the existence of a \(P_k\)-decomposition of \(\lambda K_{n,n}\) are (i) \(\lambda n^2 \equiv 0 \pmod{k-1}\) and (ii) \(k \leq n+1\) if \(\lambda=1\) and \(n\) is odd, or \(k \leq 2n\) if \(\lambda \geq 2\) or \(n\) is even. In this paper, we show these necessary conditions are sufficient except for the possibility of the case that \(k=3\), \(n=15\), and \(k=28\).

We describe a technique for producing self-dual codes over rings and fields from symmetric designs. We give special attention to biplanes and determine the minimum weights of the codes formed from these designs. We give numerous examples of self-dual codes constructed including an optimal code of length \(22\) over \(\mathbb{Z}_4\) with respect to the Hamming metric from the biplane of order \(3\).

The distance graph \(G(S, D)\) has vertex set \(V(G(S, D)) = S \cup \mathbb{R}^n\) and two vertices \(u\) and \(v\) are adjacent if and only if their distance \(d(u, v)\) is an element of the distance set \(D \subseteq \mathbb{R}_+\).

We determine the chromatic index, the choice index, the total chromatic number and the total choice number of all distance graphs \(G(\mathbb{R}, D)\), \(G(\mathbb{Q}, D)\) and \(G(\mathbb{Z}, D)\) transferring a theorem of de Bruijn and Erdős on infinite graphs. Moreover, we prove that \(|D| + 1\) is an upper bound for the chromatic number and the choice number of \(G(S,D)\), \(S \subseteq \mathbb{R}\).