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.

A join graph is the complete union of two arbitrary graphs. An edge cover coloring is a coloring of edges of \(E(G)\) such that each color appears at each vertex \(v \in V(G)\) at least one time. The maximum number of colors needed to edge cover color \(G\) is called the edge cover chromatic index of \(G\) and denoted by \(\chi’C(G)\). It is well known that any simple graph \(G\) has the edge cover chromatic index equal to \(\delta(G)\) or \(\delta(G) – 1\), where \(\delta(G)\) is the minimum degree of \(G\). If \(\chi’C(G) = \delta(G)\), then \(G\) is of C1-Class , otherwise \(G\) is of C2-Class . In this paper, we give some sufficient conditions for a join graph to be of C1-Class.

Let \(G = (V, E)\) be a simple connected graph with vertex set \(V\) and edge set \(E\). The Wiener index of \(G\) is defined by \(W(G) = \sum_{x,y \subseteq V} d(x,y),\) where \(d(x,y)\) is the length of the shortest path from \(x\) to \(y\). The Szeged index of \(G\) is defined by \(S_z(G) = \sum_{e =uv\in E} n_u(e|G) n_v(e|G),\) where \(n_u(e|G)\) (resp. \(n_v(e|G)\)) is the number of vertices of \(G\) closer to \(u\) (resp. \(v\)) than \(v\) (resp. \(u\)). The Padmakar-Ivan index of \(G\) is defined by \(PI(G) = \sum_{e =uv \in E} [n_{eu}(e|G) + n_{ev}(e|G)],\) where \(n_{eu}(e|G)\) (resp. \(n_{ev}(e|G)\)) is the number of edges of \(G\) closer to \(u\) (resp. \(v\)) than \(v\) (resp. \(u\)). In this paper, we will consider the graph of a certain nanostar dendrimer consisting of a chain of hexagons and find its topological indices such as the Wiener, Szeged, and \(PI\) index.

In this paper, we introduce a class of digraphs called \((l,m)\)-walk-regular digraphs, a common generalization of both weakly distance-regular digraphs \([1]\) and \(k\)-walk-regular digraphs \([3]\), and give several characterizations of them about their regularity properties that are related to distance and about the number of walks of given length between vertices at a given distance.