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.

In this paper, we investigate some basic properties of these eight kinds of transformation digraphs.

For any given \(k\)-uniform list assignment \(L\), a graph \(G\) is equitably \(k\)-choosable if and only if \(G\) is \(\ell\)-colorable and each color appears on at most \(\lceil \frac{|V(G)|}{k} \rceil\) vertices. A graph \(G\) is equitably \(\ell\)-colorable if \(G\) has a proper vertex coloring with \(k\) colors such that the size of the color classes differ by at most \(1\). In this paper, we prove that every planar graph \(G\) without \(6\)- and \(7\)-cycles is equitably \(k\)-colorable and equitably \(k\)-choosable where \(k \geq \max\{\Delta(G), 6\}\).

This paper introduces the concepts of forcing \(m\)-convexity number and forcing clique number of a graph. We show that the forcing \(m\)-convexity numbers of some Cartesian product and composition of graphs are related to the forcing clique numbers of the graphs. We also show that the forcing \(m\)-convexity number of the composition \(G[K_n]\), where \(G\) is a connected graph with no extreme vertex, is equal to the forcing \(m\)-convexity number of \(G\).

A spectrally arbitrary pattern \({A}\) is a sign pattern of order \(n\) such that every monic real polynomial of degree \(n\) can be achieved as the characteristic polynomial of a matrix with sign pattern \({A}\). A sign pattern \({A}\) is minimally spectrally arbitrary if it is spectrally arbitrary but is not spectrally arbitrary if any nonzero entry (or entries) of \({A}\) is replaced by zero. In this paper, we introduce some new sign patterns which are minimally spectrally arbitrary for all orders \(n\geq 7\).

Let \(G\) be a graph with vertex-set \(V = V(G)\) and edge-set \(E = E(G)\), and let \(e = |E(G)|\) and \(v = |V(G)|\). A one-to-one map \(\lambda\) from \(V \cup E\) onto the integers \(\{1, 2, \ldots, v+e\}\) is called a vertex-magic total labeling if there is a constant \(k\) so that for every vertex \(x\),

\[\lambda(x) + \sum \lambda(xy) = k\]

where the sum is over all edges \(xy\) where \(y\) is adjacent to \(x\). Let us call the sum of labels at vertex \(x\) the weight \(w_\lambda\) of the vertex under labeling \(\lambda\); we require \(w_\lambda(x) = k\) for all \(x\). The constant \(k\) is called the magic constant for \(\lambda\).

A sun \(S_n\) is a cycle on \(n\) vertices \(C_n\), for \(n \geq 3\), with an edge terminating in a vertex of degree \(1\) attached to each vertex.

In this paper, we present the vertex-magic total labeling of the union of suns, including the union of $m$ non-isomorphic suns for any positive integer $m \geq 3$, proving the conjecture given in [6].

The Randić index of an organic molecule whose molecular graph is \(G\) is the sum of the weights \((d(u)d(v))^{1/2}\) of all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of the vertex \(u\) of the molecular graph \(G\). Among all trees with \(n\) vertices and \(k\) pendant vertices, the extremal trees with the minimum, the second minimum, and the third minimum Randić index were characterized by Hansen, Li, and Wu \(et al\)., respectively. In this paper, we further investigate some small Randić index properties and give other elements of small Randić index ordering of trees with \(k\) pendant vertices.

Consider a complete graph of multiplicity \(2\), where between every pair of vertices there is one red and one blue edge. Can the edge set of such a graph be decomposed into isomorphic copies of a \(2\)-coloured path of length \(2k\) that contains \(k\) red and\(k\) blue edges? A necessary condition for this to be true is \(n(n-1) \equiv 0 \mod k\). We show that this is sufficient for \(k \leqq 3\).

In this paper, we investigate super-simple cyclic \((v, k, \lambda)\)-BIBDs (SCBIBs). Some general constructions for SCBIBs are given. The spectrum of super-simple cyclic \((v, 3, \lambda)\) is completely determined for \(\lambda = 2, 3\) and \(v – 2\). From that, some new optical orthogonal codes are obtained.