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 \(tP_4\).

For a graph \(G\), the Merrifield-Simmons index \(i(G)\) and the Hosoya index \(z(G)\) are defined as the total number of independent sets and the total number of matchings of the graph \(G\), respectively. In this paper, we characterize the graphs with the maximal Merrifield-Simmons index and the minimal Hosoya index, respectively, among the bicyclic graphs on \(n\) vertices with a given girth \(g\).

In this paper, we study the existence of \(\alpha\)-labelings for trees by means of particular \((0, 1)\)-matrices called \(a\)-labeling matrices. It is shown that each comet \(S_{k, q}\) admits no \(a\)-labelings whenever \(k > 4(q – 1)\) and \(q \geq 2\). We also give the sufficient conditions for the nonexistence of \(a\)-labelings for trees of diameter at most six. This extends a result of Rosa’s. As a consequence, we prove that \(S_{k, 3}\) has an \(a\)-labeling if and only if \(k \leq 4\).

Given a graph \(G\), an independent set \(I(G)\) is a subset of the vertices of \(G\) such that no two vertices in \(I(G)\) are adjacent. The independence number \(\alpha(G)\) is the order of a largest set of independent vertices. In this paper, we study the independence number for the Generalized Petersen graphs, finding both sharp bounds and exact results for subclasses of the Generalized Petersen graphs.