Let \(G\) be a connected simple graph. The hyper-Wiener index \(WW(G)\) is defined as \(WW(G) = \sum_{u,v \in V(G)} (d(u, v) + d^2(u,v)),\) with the summation going over all pairs of vertices in \(G\). In this paper, we determine the extremal unicyclic graphs with given matching number and minimal hyper-Wiener index.

Robertson \(([5])\) and independently, Bondy \(([1])\) proved that the generalized Petersen graph \(P(n, 2)\) is non-hamiltonian if \(n \equiv 5 \pmod{6}\), while Thomason \([7]\) proved that it has precisely \(3\) hamiltonian cycles if \(n \equiv 3 \pmod{6}\). The hamiltonian cycles in the remaining generalized Petersen graphs were enumerated by Schwenk \([6]\). In this note we give a short unified proof of these results using Grinberg’s theorem.

We present some binomial identities for sums of the bivariate Fibonacci polynomials and for weighted sums of the usual Fibonacci polynomials with indices in arithmetic progression.

Let \(v \equiv k-1, 0, \text{ or } 1 \pmod{k}\). An \(\text{RMP}(k, \lambda, v)\) (resp. \(\text{RMC}(k, \lambda, v)\)) is a resolvable packing (resp. covering) with maximum (resp. minimum) possible number \(m(v)\) of parallel classes which are mutually distinct, each parallel class consists of \(\left\lfloor \frac{v – k + 1}{k} \right\rfloor\) blocks of size \(k\) and one block of size \(v – k \left\lfloor \frac{v – k + 1}{k} \right\rfloor\), and its leave (resp. excess) is a simple graph. Such designs were first introduced by Fang and Yin. They have proved that these designs can be used to construct certain uniform designs which have been widely applied in industry, system engineering, pharmaceutics, and natural science. In this paper, direct and recursive constructions are discussed for such designs. The existence of an \(\text{RMP}(3, 3, v)\) and an \(\text{RMC}(3, 3, v)\) is proved for any admissible \(v\).

A digraph \(D\) is said to be \({super-mixed-connected}\) if every minimum general cut of \(D\) is a local cut. In this paper, we characterize non-super-mixed-connected line digraphs. As a consequence, if \(D\) is a super-arc-connected digraph with \(\delta(D) \geq 3\), then the \(n\)-th iterated line digraph of \(D\) is super-mixed-connected for any positive integer \(n\). In particular, the Kautz network \(K(d,n)\) is super-mixed-connected for \(d \neq 2\), and the de Bruijn network \(B(d,n)\) is always super-mixed-connected.

Let \(G\) be an even degree multigraph and let \(deg(v)\) and \(p(uv, G)\) denote the degree of vertex \(v\) in \(G\) and the multiplicity of edge \((u, v)\) respectively in \(G\). A decomposition of \(G\) into multigraphs \(G_1\) and \(G_2\) is said to be a \({well-spread \;halving}\) of \(G\) into two halves \(G_1\) and \(G_2\), if for each vertex \(v\), \(deg(v, G_1) = deg(v, G_2) = \frac{1}{2}deg(v, G)\), and \(|\mu(uv, G_1) – \mu(uv, G_2)| \leq 1\) for each edge \((u,v) \in E(G)\). A sufficient condition was given in \([7]\) under which there exists a well-spread halving of \(G\) if we allow the addition/removal of a Hamilton cycle to/from \(G\). Analogous to \([7]\), in this paper we define a well-spread halving of a directed multigraph \(D\) and give a sufficient condition under which there exists a well-spread halving of \(D\) if we allow the addition/removal of a particular type of Hamilton cycle to/from \(D\).

In this paper, we study linear transformations preserving log-convexity, when the triangular array satisfies some ordinary convolution. As applications, we show that the Stirling transformations of two kinds, the Lah transformation, the generalized Stirling transformation of the second kind, and the Dowling transformations of two kinds preserve the log-convexity.

For \(r \geq 3\), a \({clique-extension}\) of order \(r + 1\) is a connected graph that consists of a \(K_r\), plus another vertex adjacent to at most \(r – 1\) vertices of \(K_r\). In this paper, we consider the problem of finding the smallest number \(t\) such that any graph \(G\) of order \(n\) admits a decomposition into edge-disjoint copies of a fixed graph \(H\) and single edges with at most \(\tau\) elements. Here, we solve the case when \(H\) is a fixed clique-extension of order \(r + 1\), for all \(r \geq 3\), and will also obtain all extremal graphs. This work extends results proved by Bollobás [Math. Proc. Cambridge Philos. Soc. \(79 (1976) 19-24]\) for cliques.

A path in an edge-coloring graph \(G\), where adjacent edges may be colored the same, is called a \({rainbow\; path}\) if no two edges of \(G\) are colored the same. A nontrivial connected graph \(G\) is \({rainbow\; connected}\) if for any two vertices of \(G\) there is a rainbow path connecting them. The \({rainbow\; connection \;number}\) of \(G\), denoted \(\text{rc}(G)\), is defined as the minimum number of colors by using which there is coloring such that \(G\) is rainbow connected. In this paper, we study the rainbow connection numbers of line graphs of triangle-free graphs, and particularly, of \(2\)-connected triangle-free graphs according to their ear decompositions.

A construction based on Legendre sequences is presented for a doubly-extended binary linear code of length \(2p + 2\) and dimension \(p + 1\). This code has a double circulant structure. For \(p = 4k + 3\), we obtain a doubly-even self-dual code. Another construction is given for a class of triply extended rate \(1/3\) codes of length \(3p + 3\) and dimension \(p + 1\). For \(p = 4k + 1\), these codes are doubly-even self-orthogonal.