Let \(G\) be a nontrivial connected graph of order \(n\), and \(k\) an integer with \(2 \leq k \leq n\). For a set \(S\) of \(k\) vertices of \(G\), let \(\nu(S)\) denote the maximum number \(\ell\) of edge-disjoint trees \(T_1, T_2, \ldots, T_\ell\) in \(G\) such that \(V(T_i) \cap V(T_j) = S\) for every pair \(i, j\) of distinct integers with \(1 \leq i, j \leq \ell\). Chartrand et al. generalized the concept of connectivity as follows: The \(k\)-connectivity, denoted by \(\kappa_k(G)\), of \(G\) is defined by \(\kappa_k(G) = \min\{\nu(S)\}\), where the minimum is taken over all \(k\)-subsets \(S\) of \(V(G)\). Thus \(\kappa_2(G) = \kappa(G)\), where \(\kappa(G)\) is the connectivity of \(G\). Moreover, \(\kappa_n(G)\) is the maximum number of edge-disjoint spanning trees of \(G\).

This paper mainly focuses on the \(k\)-connectivity of complete bipartite graphs \(K_{a,b}\), where \(1 \leq a \leq b\). First, we obtain the number of edge-disjoint spanning trees of \(K_{a,b}\), which is \(\lfloor \frac{ab}{a+b-1}\rfloor \), and specifically give the \(\lfloor \frac{ab}{a+b-1}\rfloor\) edge-disjoint spanning trees. Then, based on this result, we get the \(k\)-connectivity of \(K_{a,b}\) for all \(2 \leq k \leq a + b\). Namely, if \(k > b – a + 2\) and \(a – b + k\) is odd, then \(\kappa_k(K_{a,b}) =\frac{a+b-k+1}{2} \left\lfloor \frac{(a-b + k + 1)(b-a + k – 1)}{4(k-1)} \right\rfloor\), if \(k > b – a + 2\) and \(a – b + k\) is even, then \(\kappa_k(K_{a,b}) = \frac{a+b-k+1}{2} +\left\lceil \frac{(a – b+ k )(a + b – k)}{4(k-1)} \right\rceil\), and if \(k \leq b – a + 2\), then \(\kappa_k(K_{a,b}) = a\).

A labelling of a graph over a field \(\mathbb{F}\) is a mapping of the edge set of the graph into \(\mathbb{F}\). A labelling is called magic if for any vertex, the sum of the labels of all the edges incident to it is the same. The class of all such labellings forms a vector space over \(\mathbb{F}\) and is called the magic space of the graph. For finite graphs, the dimensional structure of the magic space is well known. In this paper, we give the existence of magic labellings and discuss the dimensional structure of the magic space of locally finite graphs. In particular, for a class of locally finite graphs, we give an explicit basis of the magic space.

For two positive integers \(j\) and \(k\) with \(j \geq k\), an \(L(j,k)\)-labeling of a graph \(G\) is an assignment of nonnegative integers to \(V(G)\) such that the difference between labels of adjacent vertices is at least \(j\), and the difference between labels of vertices that are distance two apart is at least \(k\). The span of an \(L(j, k)\)-labeling of a graph \(G\) is the difference between the maximum and minimum integers used by it. The \(\lambda_{j,k}\)-number of \(G\) is the minimum span over all \(L(j, k)\)-labelings of \(G\). This paper focuses on the \(\lambda_{2,1}\)-number of the Cartesian products of complete graphs. We completely determine the \(\lambda_{2,1}\)-numbers of the Cartesian products of three complete graphs \(K_n\), \(K_m\), and \(K_l\): for any three positive integers \(n\), \(m\), and \(l\).

Let \(G = (V(G), E(G))\) be a graph. A set \(S \subseteq V(G)\) is a packing if for any two vertices \(u\) and \(v\) in \(S\) we have \(d(u, v) \geq 3 \). That is, \(S\) is a packing if and only if for any vertex \(v \in V(G)\), \(|N[v] \cap S| \leq 1\). The packing number \(\rho(G)\) is the maximum cardinality of a packing in \(G\). In this paper, we study the packing number of generalized Petersen graphs \(P(n,2)\) and prove that \(\rho(P(n,2)) = \left\lfloor \frac{n}{7} \right\rfloor + \left\lceil \frac{n+1}{7} \right\rceil + \left\lfloor \frac{n+4}{7} \right\rfloor\) (\(n \geq 5\)).

Let \(G\) be a connected graph. The Wiener index of \(G\) 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 summation goes over all the unordered pairs of vertices. In this paper, we investigate the Wiener index of unicyclic graphs with given girth and characterize the extremal graphs with the second maximal and second minimal Wiener index.

This paper uses research methods in the subspace lattices, making a deep research to the lattices of all subsets of a finite set and partition of an n-set. At first, the inclusion relations between different lattices are studied. Then, a characterization of elements contained in a given lattice is given. Finally, the characteristic polynomials of the given lattices are computed.

Let \(G\) be a connected graph on \(n\) vertices. The average eccentricity of a graph \(G\) is defined as \(\varepsilon(G) = \frac{1}{n} \sum_{v \in V(G)} \varepsilon(v)\), where \(\varepsilon(v)\) is the eccentricity of the vertex \(v\), which is the maximum distance from it to any other vertex. In this paper, we characterize the extremal unicyclic graphs among \(n\)-vertex unicyclic graphs having the minimal and the second minimal average eccentricity.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A (defensive) alliance in \(G\) is a subset \(S\) of \(V(G)\) such that for every vertex \(v \in S\), \(|N(v) \cap S| \geq |N(v) \cap (V(G) – S)|\). The alliance partition number of a graph \(G\), \(\psi_a(G)\), is defined to be the maximum number of sets in a partition of \(V(G)\) such that each set is a (defensive) alliance. In this paper, we give both general bounds and exact results for the alliance partition number of graphs, and in particular for regular graphs and trees.

In this paper, we present a unified and simple approach to extremal acyclic graphs without perfect matching for the energy, the Merrifield-Simmons index and Hosoya index.

The notion of equitable coloring was introduced by Meyer in \(1973\). In this paper, we obtain interesting results regarding the equitable chromatic number \(\chi=\) for the sun let graphs \(S_n\), line graph of sun let graphs \(L(S_n)\), middle graph of sun let graphs \(M(S_n)\), and total graph of sun let graphs \(T(S_n)\).