The hyper Wiener index of a connected graph \(G\) is defined as
\(WW(G) = \frac{1}{2}\sum_{u,v \in V(G)} d(u,v) + \frac{1}{2}\sum_{(u,v) \in V(G)} d(u,v)^2\) where \(d(u, v)\) is the distance between vertices \(u,v \in V(G)\).
In this paper we find an exact expression for hyper Wiener index of \(HC_6[p, q]\), the zigzag polyhex nanotori.

In this paper, we classify all optimal linear \([n, n/2]\) codes over \(\mathbb{Z}_4\) up to length \(n = 8\), and determine the number of optimal codes which are self-dual and formally self-dual. Optimal codes with linear binary images are identified. In particular, we show that for length \(8\), there are nine optimal codes for the Hamming distance, one optimal code for the Lee distance, and two optimal codes for the Euclidean distance.

In this paper, we show that if \(k \geq \frac{v+2}{4}\), where \(v\) denotes the order of a graph, a non-bipartite graph \(G\) is \(k\)-extendable if and only if it is \(2k\)-factor-critical. If \(k \geq \frac{v-3}{4}\), a graph \(G\) is \(k\)-extendable if and only if it is \((2k+1)\)-factor-critical. We also give examples to show that the two bounds are best possible. Our results are answers to a problem posted by Favaron \([3]\) and Yu \([11]\).

The edge-neighbor-scattering number of a graph \(G\) is defined to be \(EN_S(G) = \max\limits_{S\subseteq E(G)}\{w(G/S) -\mid |S|\}\) where \(S\) is any edge-cut-strategy of \(G\), \(w(G/S)\) is the number of the components of \(G/S\). In this paper, we give edge-neighbor-scattering number of some special classes of graphs, and then mainly discuss the general properties of the parameter.

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.