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.

The Merrifield-Simmons index \(\sigma(G)\) of a (molecular) graph \(G\) is defined as the number of independent-vertex sets of \(G\). By \(G(n, l, k)\) we denote the set of unicyclic graphs with girth \(l\) and the number of pendent vertices being \(k\) respectively. Let \(S_n^l\) be the graph obtained by identifying the center of the star \(S_{n-l+1}\) with any vertex of \(C_l\). By \(S^{l,k}_n*\) we denote the graph obtained by identifying one pendent vertex of the path \(P_{n-l-k+1}\) with one pendent vertex of \(S_{l+k}^l\). In this paper, we first investigate the Merrifield-Simmons index for all unicyclic graphs in \(G(n,l,k)\) and \(S^{l,k}_n*\) is shown to be the unique unicyclic graph with maximum Merrifield-Simmons index among all unicyclic graphs in \(G(n, l, k)\) for fixed \(l\) and \(k\). Moreover, we proved that:

1. When \(k = n – 3\), \(S^{3,k}_n\) has the maximum Merrifield-Simmons index among all graphs in \(G(n, k)\); When \(k = 1, n-4\), \(S^{4,k}_n\) or \(S^{n-k,k}_n\) has the maximum Merrifield-Simmons index among all graphs in \(G(n,k)\)
2. When \(2 \leq k \leq n-5\), \(S^{n-k,k}_n\) and \(S^{4,k}_n\) are respectively unicyclic graphs having maximum and second-maximum Merrifield-Simmons indices among all unicyclic graphs in \(G(n, k)\), where \(G(n, k)\) denotes the set of unicyclic graphs with \(n\) vertices and \(k\) pendent vertices.

In this paper, we give a complete solution to the Hamilton-Waterloo problem for the case of Hamilton cycles and \(C_{4k}\)-factors for all positive integers \(k\).

In this paper, we study the edge deletion preserving the diameter of the Johnson graph \(J(n,k)\). Let \(un^-(G)\) be the maximum number of edges of a graph \(G\) whose removal maintains its diameter. For Johnson graph \(J(n,k)\), we give upper and lower bounds to the number \(un^-(J(n,k))\), namely:\(\binom{k}{2}\binom{n}{k+1} \leq un^-(J(n,k)) \leq \binom{k+1}{2} \binom{n}{k+1} + \lceil(1+\frac{1}{2k})(\binom{n}{k} – 1\rceil,\) for \(n \geq 2k \geq 2\).

In this paper, we study the global behavior of the nonnegative equilibrium points of the difference equation

\[x_{n+1} = \frac{ax_{n-k}}{bcx_{n-k}^rx_{n-(2k+1)}^s}, \quad n=0,1,\ldots\]

where \(a, b, c, d, e\) are nonnegative parameters, initial conditions are nonnegative real numbers, \(k\) is a nonnegative integer, and \(r, s \geq 1\).

Let \(\mathcal{I}_X\) be the symmetric inverse semigroup on a finite nonempty set \(X\), and let \(A\) be a subset of \(\mathcal{I}^*_X = \mathcal{I}_X \setminus \{0\}\). Let \(\text{Cay}(\mathcal{I}^*_X, A)\) be the graph obtained by deleting vertex \(0\) from the Cayley graph \(\text{Cay}(\mathcal{I}_X, A)\). We obtain conditions on \(\text{Cay}(\mathcal{I}^*_X, A)\) for it to be \(\text{ColAut}_A(\mathcal{I}^*_X)\)-vertex-transitive and \(\text{Aut}_A(\mathcal{I}^*_X)\)-vertex-transitive. The basic structure of vertex-transitive \(\text{Cay}(\mathcal{I}^*_X, A)\) is characterized. We also investigate the undirected Cayley graphs of symmetric inverse semigroups, and prove that the generalized Petersen graph can be constructed as a connected component of a Cayley graph of a symmetric inverse semigroup, by choosing an appropriate connecting set.