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.

A cograph is a \(P_4\)-free graph. We first give a short proof of the fact that \(0\) (\(-1\)) belongs to the spectrum of a connected cograph (with at least two vertices) if and only if it contains duplicate (resp. coduplicate) vertices. As a consequence, we next prove that the polynomial reconstruction of graphs whose vertex-deleted subgraphs have the second largest eigenvalue not exceeding \(\frac{\sqrt{5}-1}{2}\) is unique.

In this paper, we describe Cayley graphs of rectangular bands and normal bands, which are the strong semilattice of rectangular bands, respectively. In particular, we give the structure of Cayley graphs of rectangular bands and normal bands, and we determine which graphs are Cayley graphs of rectangular bands and normal bands.

The generalized Petersen graph \(P(n, k)\) is the graph whose vertex set is \(U \cup W\), where \(U = \{u_0, u_1, \ldots, u_{n-1}\}\), \(W = \{v_0, v_1, \ldots, v_{n-1}\}\); and whose edge set is \(\{u_iu_{i+1},u_iv_{i}, v_iv_{i+k} \mid i = 0, 1, \ldots, n-1\}\), where \(n, k\) are positive integers, addition is modulo \(n\), and \(2 < k < n/2\). G. Exoo, F. Harary, and J. Kabell have determined the crossing number of \(P(n, 2)\); Richter and Salazar have determined the crossing number of the generalized Petersen graph \(P(n, 3)\). In this paper, the crossing number of the generalized Petersen graph \(P(3k, k)\) (\(k \geq 4\)) is studied, and it is proved that \(\text{cr}(P(3k,k)) = k\) (\(k \geq 4\)).

In this paper, we apply the concept of fundamental relation on \(\Gamma\)-hyperrings and obtain some related results. Specially, we show that there is a covariant functor between the category of \(\Gamma\)-hyperrings and the category of fundamental \(\Gamma’/\beta^*\)-rings.