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 \(T(S_n)\).

Kühn and Osthus \([2]\) proved that for every positive integer \(\ell\), there exists an integer \(k(\ell) \leq 2^{11}.3\ell^2\), such that the vertex set of every graph \(G\) with \(\delta(G) \geq k(\ell)\) can be partitioned into subsets \(S\) and \(T\) with the properties that \(\delta(G[S]) \geq \ell \leq \delta(G[T])\) and every vertex of \(S\) has at least \(\ell\) neighbors in \(T\). In this note, we improve the upper bound to \(k(\ell) \leq 2^4 – 17\ell^2\).

In this paper, we discuss how the addition of a new edge changes the irregularity strength in \(K(3,n)\), \(tK_3\), and \(tP_4\).