We construct a record-breaking binary code of length \(17\), minimal distance \(6\), constant weight \(6\), and containing \(113\) codewords.

The purpose of this note is to give the power formula of the generalized Lah matrix and show \(\mathcal{L}[x,y] = \mathcal{FQ}[x,y]\), where \(\mathcal{F}\) is the Fibonacci matrix and \(\mathcal{Q}[x,y]\) is the lower triangular matrix. From it, several combinatorial identities involving the Fibonacci numbers are obtained.

A graph is called set reconstructible if it is determined uniquely (up to isomorphism) by the set of its vertex-deleted subgraphs. We prove that some classes of separable graphs with a unique endvertex are set reconstructible and show that all graphs are set reconstructible if all \(2\)-connected graphs are set reconstructible.

We prove the following extension of the Erdős-Ginzburg-Ziv Theorem. Let \(m\) be a positive integer. For every sequence \(\{a_i\}_{i\in I}\) of elements from the cyclic group \(\mathbb{Z}_m\), where \(|I| = 4m – 5\) (where \(|I| = 4m – 3\)), there exist two subsets \(A, B \subseteq I\) such that \(|A \cap B| = 2\) (such that \(|A \cap B| = 1\)), \(|A| = |B| = m\), and \(\sum\limits_{i\in b} a_i = \sum\limits_{i\in b} b_i = 0\).

A connected graph is said to be super edge-connected if every minimum edge-cut isolates a vertex. The restricted edge-connectivity \(\lambda’\) of a connected graph is the minimum number of edges whose deletion results in a disconnected graph such that each component has at least two vertices. It has been shown by A. H. Esfahanian and S. L. Hakimi (On computing a conditional edge-connectivity of a graph. Information Processing Letters, 27(1988), 195-199] that \(\lambda'(G) \leq \xi(G)\) for any graph of order at least four that is not a star, where \(\xi(G) = \min\{d_G(u) + d_G(v) – 2: uv \text{ is an edge in } G\}\). A graph \(G\) is called \(\lambda’\)-optimal if \(\lambda'(G) = \xi(G)\). This paper proves that the de Bruijn undirected graph \(UB(d,n)\) is \(\lambda’\)-optimal except \(UB(2,1)\), \(UB(3,1)\), and \(UB(2,3)\), and hence, is super edge-connected for \(n\geq 1\) and \(d\geq 2\).

The problem of graceful labeling of a particular class of trees called quasistars is considered. Such a quasistar is a tree \(Q\) with \(k\) distinct paths with lengths \(1, d+1, 2d+1, \ldots, (k-1)d+1\) joined at a unique vertex \(\theta\).

Thus, \(Q\) has \(1 + [1 + (d+1) + (2d+1) + \ldots + (k-1)d+1)] = 1+k +\frac{k(k-1)d}{2}\) vertices. The \(k\) paths of \(Q\) have lengths in arithmetic progression with common difference \(d\). It is shown that \(Q\) has a graceful labeling for all \(k \leq 6\) and all values of \(d\).

The average distance \(\mu(D)\) of a strong digraph \(D\) is the average of the distances between all ordered pairs of distinct vertices of \(D\). Plesnik \([3]\) proved that if \(D\) is a strong tournament of order \(n\), then \(\mu(D) \leq \frac{n+4}{6} + \frac{1}{n}\). In this paper we show that, asymptotically, the same inequality holds for strong bipartite tournaments. We also give an improved upper bound on the average distance of a \(k\)-connected bipartite tournament.

To measure the efficiency of a routing in network, Chung et al. [The forwarding index of communication networks. IEEE Trans. Inform. Theory, 33 (2) (1987), 224-232] proposed the notion of forwarding index and established an upper bound \((n – 1)(n – 2)\) of this parameter for a connected graph of order \(n\). This note improves this bound as \((n – 1)(n – 2) – (2n – 2 – \Delta\lfloor1+\frac{n-1}{\Delta}\rfloor)\) \(\lfloor \frac{n-1}{\Delta}\rfloor\) , where \(\Delta\) is the maximum degree of the graph \(G\). This bound is best possible in the sense that there is a graph \(G\) attaining it.

We study the spectral radius of unicyclic graphs with \(n\) vertices and edge independence number \(q\). In this paper, we show that of all unicyclic graphs with \(n\) vertices and edge independence number \(q\), the maximal spectral radius is obtained uniquely at \(\Delta_n(q)\), where \(\Delta_n(q)\) is a graph on \(n\) vertices obtained from the cycle \(C_3\) by attaching \(n – 2q + 1\) pendant edges and \(q – 2\) paths of length \(2\) at one vertex.

Let \(q\) be an odd prime power and \(p\) be an odd prime with \(gcd(p, g) = 1\). Let the order of \(g\) modulo \(p\) be \(f\) and \(gcd(\frac{p-1}{f}, q) = 1\). Here explicit expressions for all the primitive idempotents in the ring \(R_{2p^n} = GF(q)[x]/(x^{2p^n} – 1)\), for any positive integer \(n\), are obtained in terms of cyclotomic numbers, provided \(p\) does not divide \(\frac{q^f-1}{2p}\), if \(n \geq 2\). Some lower bounds on the minimum distances of irreducible cyclic codes of length \(2p^n\) over \(GF(q)\) are also obtained.