We develop the necessary machinery in order to prove that hexagonal tilings are uniquely determined by their Tutte polynomial, showing as an example how to apply this technique to the toroidal hexagonal tiling.

A \((d,1)\)-totel labelling of a graph \(G\) is an assignment of integers to \(V(G) \cap E(G)\) such that: (i) any two adjacent vertices of \(G\) receive distinct integers, (ii) any two adjacent edges of \(G\) receive distinct integers, and (iii) a vertex and its incident edge receive integers that differ by at least \(d\) in absolute value. The span of a \((d,1)\)-total labelling is the maximum difference between two labels. The minimum span of labels required for such a \((d, 1)\)-total labelling of \(G\) is called the \((d, 1)\)-total number and is denoted by \(\lambda_d^T(G)\). In this paper, we prove that \(\lambda_d^T(G)\geq d+r+1 \) for \(r\)-regular nonbipartite graphs with \(d \geq r \geq 3\) and determine the \((d, 1)\)-total numbers of flower snarks and of quasi flower snarks.

Let \(G = (V,E)\) be a simple graph with the vertex set \(V\) and the edge set \(E\). \(G\) is a sum graph if there exists a labelling \(f\) of the vertices of \(G\) into distinct positive integers such that \(uv \in E\) if and only if \( f(w)=f(u) + f(v) \) for some vertex \(w \in V\). Such a labelling \(f\) is called a sum labelling of \(G\). The sum number \(\sigma(G)\) of \(G\) is the smallest number of isolated vertices which result in a sum graph when added to \(G\). Similarly, the integral sum graph and the integral sum number \(\zeta(G)\) are also defined. The difference is that the labels may be any distinct integers.
In this paper, we will determine that
\[\begin{cases}
0 = \zeta(\overline{P_4}) < \sigma(\overline{P_4}) = 1;\\ 1 = \zeta(\overline{P_5}) < \sigma(\overline{P_5}) = 2;\\ 3 = \zeta(\overline{P_6}) < \sigma(\overline{P_6}) = 4;\\ \zeta(\overline{P_n}) = \sigma(\overline{P_n}) = 0, \text{ for } n = 1, 2, 3;\\ \zeta(\overline{P_n}) = \sigma(\overline{P_n}) = 2n – 7, \text{ for } n \geq 7; \end{cases}\] and \[\begin{cases} 0 = \zeta(\overline{F_5}) < \sigma(\overline{F_5}) = 1;\\ 2 = \zeta(\overline{F_5}) < \sigma(\overline{F_6}) = 2;\\ \zeta(\overline{F_c}) = \sigma(\overline{F_n}) = 0, \text{ for } n =3,4;\\ \zeta(\overline{F_n}) = \sigma(\overline{F_n}) = 2n – 8, \text{ for } n \geq 7. \end{cases}\]

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the PI indices of bicyclic graphs whose cycles do not share two or more common vertices.

For each of the parameter sets \((30, 7, 15)\) and \((26, 12, 55)\), a simple \(3\)-design is given. They have \(\text{PSL}(2, 29)\) and \(\text{PSL}(2, 25)\) as their automorphism group, respectively. Each of the two simple \(3\)-designs is the first one ever known with the parameter set given and \(4\) in each of the two parameter sets is minimal for the given \(v\) and \(k\).

In this paper, we study linear codes over finite chain rings. We relate linear cyclic codes, \((1 + \gamma^k)\)-cyclic codes and \((1 – \gamma^k)\)-cyclic codes over a finite chain ring \(R\), where \(\gamma\) is a fixed generator of the unique maximal ideal of the finite chain ring \(R\), and the nilpotency index of \(\gamma\) is \(k+1\). We also characterize the structure of \((1+\gamma^k)\)-cyclic codes and \((1 – \gamma^k)\)-cyclic codes over finite chain rings.

Let \(G\) be a graph with \(n\) vertices. The mean integrity of \(G\) is defined as follows:\(J(G) = min_{P \subseteq V} \{|P| + \tilde{m}(G – P)\},\) where \(\tilde{m}(G – P) = \frac{1}{n-|P|}\sum_{v \in G – P} n_v\) and \(n_v\) is the size of the component containing \(v\). The main result of this article is a formula for the mean integrity of a path \(P_n\) of \(n\) vertices. A corollary of this formula establishes the mean integrity of a cycle \(C_n\) of \(n\) vertices.

It is known that any reducible additive hereditary graph property has infinitely many minimal forbidden graphs, however the proof of this fact is not constructive. The purpose of this paper is to construct infinite families of minimal forbidden graphs for some classes of reducible properties. The well-known Hajós’ construction is generalized and some of its applications are presented.

In this paper, we prove that for any graph \(G\), there is a dominating induced subgraph which is a cograph. Two new domination parameters \(\gamma_{cd}\) – the cographic domination number and \(\gamma_{gcd}\) – the global cographic domination number are defined. Some properties, including complexity aspects, are discussed.

Given a permutation \(\pi\) chosen uniformly from \(S_n\), we explore the joint distribution of \(\pi(1)\) and the number of descents in \(\pi\). We obtain a formula for the number of permutations with \(Des(\pi) = d\) and \(\pi(1) = k\), and use it to show that if \(Des(\pi)\) is fixed at \(d\), then the expected value of \(\pi(1)\) is \(d+1\). We go on to derive generating functions for the joint distribution, show that it is unimodal if viewed correctly, and show that when \(d\) is small the distribution of \(\pi(1)\) among the permutations with \(d\) descents is approximately geometric. Applications to Stein’s method and the Neggers-Stanley problem are presented.