In this paper, we prove that for any positive integers \(k,n\) with \(k \geq 2\) , the graph \(P_k^n\) is a divisor graph if and only if \(n \leq 2k + 2\) , where \(P^k_n\) is the \(k\) th power of the path \(P_n\). For powers of cycles we show that \(C^k_n\) is a divisor graph when \(n \leq 2k + 2\), but is not a divisor graph when \(n \geq 2k + 2\),but is not a divisor graph when \(n\geq 2k+\lfloor \frac{k}{2}\rceil,\) where \(C^k_n\) is the \(k\)th power of the cycle \(C_n\). Moreover, for odd \(n\) with \(2k+2 < n < 2k + \lfloor\frac{k}{2}\rfloor + 3\), we show that the graph \(C^k_n\) is not a divisor graph.

The Wiener index of a graph \(G\) is defined as \(W(G) = \sum_{u,v \in V(G)} d_G(u,v),\) where \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\) and the sum goes over all pairs of vertices. In this paper, we investigate the Wiener index of unicyclic graphs with given girth and characterize the extremal graphs with the minimal and maximal Wiener index.

In this paper, we consider a random mapping, \(\hat{T}_n\), of the finite set \(\{1,2,\ldots,n\}\) into itself for which the digraph representation \(\hat{G}_n\) is constructed by:\((1)\) selecting a random number, \(\hat{L}_n\), of cyclic vertices,\((2)\) constructing a uniform random forest of size \(n\) with the selected cyclic vertices as roots, and \((3)\) forming `cycles’ of trees by applying a random permutation to the selected cyclic vertices.We investigate \(\hat{k}_n\), the size of a `typical’ component of \(\hat{G}_n\), and, under the assumption that the random permutation on the cyclical vertices is uniform, we obtain the asymptotic distribution of \(k\), conditioned on \(\hat{L}_n = m(n)\). As an application of our results, we show in Section \(3\) that provided \(\hat{L}_n\) is of order much larger than \(\sqrt{n}\), then the joint distribution of the normalized order statistics of the component sizes of \(\hat{G}_n\) converges to the Poisson-Dirichlet \((1)\) distribution as \(n \to \infty\). Other applications and generalizations are also discussed in Section \(3\).

De Bruijn digraphs and shuffle-exchange graphs are useful models for interconnection networks. They can be represented as group action graphs of the wrapped butterfly graph and the cube-connected cycles, respectively. The Kautz digraph has similar definitions and properties to de Bruijn digraphs. It is \(d\)-regular and strongly \(d\)-connected, thus it is a group action graph. In this paper, we use another representation of the Kautz digraph and settle the open problem posed by M.-C. Heydemann in \([6]\).

We discuss the use of \(K\)-terminal networks to represent arbitrary clutters. A given clutter has many different representations, and there does not seem to be any set of simple transformations that can be used to transform one representation of a clutter into any other. We observe that for \(t \geq 2\) the class of clutters that can be represented using no more than \(t\) terminals is closed under minors, and has infinitely many forbidden minors.

Brenti (J. Combin. Theory Ser. A \(91 (2000))\) considered a \(q\)-analogue of the Eulerian polynomials by enumerating permutations in the symmetric group \(S_n\) with respect to the numbers of excedances and cycles. Here we establish a connection between these \(q\)-Eulerian polynomials and some infinite generating functions.

Let \(K_k, C_k, T_k\), and \(P_k\) denote a complete graph on \(k\) vertices, a cycle on \(k\) vertices, a tree on \(k+1\) vertices, and a path on \(k+1\) vertices, respectively. Let \(K_m-H\) be the graph obtained from \(K_m\) by removing the edges set \(E(H)\) of the graph \(H\) (\(H\) is a subgraph of \(K_m\)). A sequence \(S\) is potentially \(K_m-H\)-graphical if it has a realization containing a \(K_m-H\) as a subgraph. Let \(\sigma(K_m-H,n)\) denote the smallest degree sum such that every \(n\)-term graphical sequence \(S\) with \(\sigma(S) \geq \sigma(K_m-H,n)\) is potentially \(K_m-H\)-graphical. In this paper, we determine the values of \(\sigma(K_{r+1}-H,n)\) for \(n \geq 4r+10, r \geq 3, r+1 \geq k \geq 4\) where \(H\) is a graph on \(k\) vertices which contains a tree on \(4\) vertices but not contains a cycle on \(3\) vertices. We also determine the values of \(\sigma(K_{r+1}-P_{2},n)\) for \(n \geq 4r+8, r \geq 3\).

In this paper, the class of \((m,n)\)-ary hypermodules is introduced and several properties and examples are found. \((m,n)\)-ary hypermodules are a generalization of hypermodules. On the other hand, we can consider \((m,n)\)-ary hypermodules as a good generalization of \((m,n)\)-ary modules. We define the fundamental relation \(\epsilon^*\) on the \((m,n)\)-ary hypermodules \(M\) as the smallest equivalence relation such that \(M/\epsilon^*\) is an \((m,n)\)-ary module, and then some related properties are investigated.

For given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1, G_2)\) is defined to be the least positive integer \(n\) such that every graph \(G\) on \(n\) vertices, either \(G\) contains a copy of \(G_1\) or the complement of \(G\) contains a copy of \(G_2\). In this note, we show that \(R(C_m, B_n) = 2m-1\) for \(m \geq 2n-1 \geq 7\). With the help of computers, we obtain the exact values of \(14\) small cycle-book Ramsey numbers.

For positive integers \(c \geq 0\) and \(k \geq 1\), let \(n = R(c, k)\) be the least integer, provided it exists, such that every \(2\)-coloring of the set \([1,n] = \{1,\ldots,n\}\) admits a monochromatic solution to the equation \(x + y+c = 4z\) with \(x, y, z \in [1,n]\). In this paper, the precise value of \(R(c, 4)\) is shown to be \(\left\lceil{3c + 2}/{8}\right\rceil\) for all even \(c \geq 34\).