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.

In this note, we show that the variety of Boolean \(SQS\)-skeins can be defined by a single axiom and, in the process, we find all of the shortest single axioms for said variety. Our investigations were aided by the automated theorem-prover Prover9 and the finite model-finder Mace4.

Let \(G(V,E)\) be a graph. A subset \(S\) of \(V\) is called a dominating set of \(G\) if every vertex in \(V-S\) is adjacent to at least one vertex in \(S\). The domination number \(\gamma(G)\) of \(G\) is the minimum cardinality taken over all dominating sets in \(G\). A dominating set \(S\) of \(G\) is called a complementary perfect dominating set (cpd-set) if the induced subgraph \(\langle V-S \rangle\) has a perfect matching. The complementary perfect domination number, \(\gamma_{cp}(G)\), of \(G\) is the minimum cardinality taken over all cpd-sets in \(G\).

An induced complementary perfect dominating set of a graph (icpd-set) is a dominating set of \(G\) such that the induced subgraph \(\langle V-S \rangle\) has only independent edges. That is, \(\langle V-S \rangle = mK_2\), \(m \geq 1\). The minimum cardinality taken over all such icpd-sets of \(G\) is called the induced complementary perfect domination number of \(G\), and is denoted by \(\gamma_{icp}(G)\).

A subset \(S\) of \(V\) is said to be a complementary connected dominating set (ccd-set) if \(S\) is a dominating set and \(\langle V-S \rangle\) is connected. The complementary connected domination number of a graph is denoted by \(\gamma_{cc}(G)\) and is defined as the minimum number of vertices which form a ccd-set.

It has been proved that \(\gamma_{cp}(G) = n = \gamma_{icp}(G)\) and \(\gamma_{cc}(G) = n-1\) only if \(G\) is a star. And if \(G\) is not a star, then \(\gamma_{cp}, \gamma_{icp}, \gamma_{cc} \leq n-2\). In this paper, we characterize the graphs with \(\gamma_{cc} \leq n-2\), and trees with \(\gamma_{cp} = n-2\) and \(\gamma_{icp} = n-2\).

A graph \(G\) is called \(H\)-equipackable if every maximal \(H\)-packing in \(G\) is also a maximum \(H\)-packing in \(G\). In 2009, \(P_4\)-equipackable paths and cycles, \(M_3\)-equipackable paths and cycles have been characterized. In this paper, \(P_k\)-equipackable paths and cycles, \(M_k\)-equipackable paths and cycles are characterized.

We determine the maximum Wiener index of \(n\)-vertex unicyclic graphs with fixed maximum degree and characterize the unique extremal graph.

The aim of this paper is to define different types of continuities of operators and boundedness of linear operators over fuzzy \(n\)-normed linear spaces. Also, some definitions such as fuzzy continuity, sequential fuzzy continuity, weakly fuzzy continuity, strongly fuzzy continuity, weakly fuzzy boundedness, and strongly fuzzy boundedness are given in fuzzy \(n\)-normed linear spaces. In addition, some theorems related to these definitions are proved.

In this paper, we study the enumeration of noncrossing partitions with fixed points. The expressions of \({f_m}(x_1, x_2,x_3, 0, \ldots, 0)\) and \({f_m}(x_1, x_2, 0, \ldots, 0, x_{p+3}, 0, \ldots, 0)\) are found, and a new proof of the expression of \({f_m}(x_1, x_2,0, 0, \ldots, 0)\) is obtained using diophantine equations.

Let \(G\) be a subgraph of \(K_n\). The graph obtained from \(G\) by replacing each edge with a 3-cycle whose third vertex is distinct from other vertices in the configuration is called a \(T(G)\)-triple. An edge-disjoint decomposition of \(3K_n\) into copies of \(T(G)\) is called a \(T(G)\)-triple system of order \(n\). If, in each copy of \(T(G)\) in a \(T(G)\)-triple system, one edge is taken from each 3-cycle (chosen so that these edges form a copy of \(G\)) in such a way that the resulting copies of \(G\) form an edge-disjoint decomposition of \(K_n\), then the \(T(G)\)-triple system is said to be perfect. The set of positive integers \(n\) for which a perfect \(T(G)\)-triple system exists is called its spectrum. Earlier papers by authors including Billington, Lindner, Kıvcıkgızı, and Rosa determined the spectra for cases where \(G\) is any subgraph of \(K_4\). In this paper, we will focus on the star graph \(K_{1,k}\) and discuss the existence of perfect \(T(K_{1,k})\)-triple systems. Especially, for prime powers \(k\), its spectra are completely determined.