Let \(M = \{v_1, v_2, \ldots, v_t\}\) be an ordered set of vertices in a graph \(G\). Then \((d(u, v_1), d(u, v_2), \ldots, d(u, v_\ell))\) is called the \(M\)-location of a vertex \(u\) of \(G\). The set \(M\) is called a locating set if the vertices of \(G\) have distinct \(M\)-locations. A minimum locating set is a set \(M\) with minimum cardinality. The cardinality of a minimum locating set of \(G\) is called the Location Number \(L(G)\). This concept has wide applications in motion planning and in the field of robotics. In this paper, we consider networks with a binary tree as an underlying structure and determine the minimum locating set of such architectures. We show that the location number of an \(n\)-level \(X\)-tree lies between \(2^{n-3}\) and \(2^{n – 3} + 2\). We further prove that the location number of an \(N \times N\) mesh of trees is greater than or equal to \(N/2\) and less than or equal to \(N\).

In this paper, we give generalizations of Padovan numbers and Perrin numbers. We apply these generalizations for counting of special subsets of the set of \(n\) integers. Next, we give their graph representations with respect to the number of maximal \(k\)-independent sets in graphs.

In this paper, we show that the crossing number of the complete multipartite graph \(K_{1,1,3,n}\) is

\[\operatorname{cr}(K_{1,1,3,n}) = 4\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor + \lfloor\frac{3n}{2}\rfloor\]

Our proof depends on Kleitman’s results for the complete bipartite graphs [D. J. Kleitman, The crossing number of \(K_{5,n}\), J. Combin.Theory, \(9 (1970), 315-323\)]..

A near-perfect matching is a matching saturating all but one vertex in a graph. In this note, it is proved that if a graph has a near-perfect matching then it has at least two, moreover, a concise structure construction for all graphs with exactly two near-perfect matchings is given. We also prove that every connected claw-free graph \(G\) of odd order \(n\) (\(n \geq 3\)) has at least \(\frac{n+1}{2}\) near-perfect matchings which miss different vertices of \(G\).

le of an edge-coloured graph \(G^*\) such that there is no finite integer \(n\) for which it is possible to decompose \(rK_n^*\) into edge-disjoint colour-identical copies of \(G^*\). We investigate the problem of determining precisely when an edge-coloured graph \(G^*\) with \(r\) colours admits a \(G^*\)-decomposition of \(rK_n^*\), for some finite \(n\). We also investigate conditions under which any partial edge-coloured \(G^*\)-decomposition of \(rK_n^*\) has a finite embedding.

Four new combinatorial identities involving certain generalized \(F\)-partition functions and \(n\)-colour partition functions are proved bijectively. This leads to new combinatorial interpretations of four mock theta functions of S.Ramanujan.

In this paper, we introduce some contractive conditions of Meir-Keeler type for a pair of mappings, called MK-pair and L-pair, in the framework of cone metric spaces. We prove theorems which assure the existence and uniqueness of common fixed points for MK-pairs and L-pairs. As an application, we obtain a result on the common fixed point of a p-MK-pair, a mapping, and a multifunction in complete cone metric spaces. These results extend and generalize well-known comparable results in the literature.

A digraph \(D(V, E)\) is said to be graceful if there exists an injection \(f : V(D) \rightarrow \{0, 1, \ldots, |V|\}\) such that the induced function \(f’ : E(D) \rightarrow \{1, 2, \ldots, |V|\}\) which is defined by \(f'(u,v) = [f(v) – f(u)] \pmod{|E| + 1}\) for every directed edge \((u,v)\) is a bijection. Here, \(f\) is called a graceful labeling (graceful numbering) of digraph \(D(V, E)\), while \(f’\) is called the induced edge’s graceful labeling of digraph \(D(V,E)\). In this paper, we discuss the gracefulness of the digraph \(n-\vec{C}_m\) and prove that the digraph \(n-\vec{C}_{17}\) is graceful for even \(n\).

Let \(G\) be a connected graph, and let \(d(u,v)\) denote the distance between vertices \(u\) and \(v\) in \(G\). For any cyclic ordering \(\pi\) of \(V(G)\), let \(\pi = (v_1, v_2, \ldots, v_n, v_{n+1} = v_1)\), and let \(d(\pi) = \sum\limits_{i=1}^n d(v_i, v_{i+1})\). The set of possible values of \(d(\pi)\) of all cyclic orderings \(\pi\) of \(V(G)\) is called the Hamiltonian spectrum of \(G\). We determine the Hamiltonian spectrum for any tree.

Candelabra quadruple systems, which are usually denoted by \(\text{CQS}(g^n : s)\), can be used in recursive constructions to build Steiner quadruple systems. In this paper, we introduce some necessary conditions for the existence of a \(\text{CQS}(g^n : s)\) and settle the existence when \(n = 4,5\) and \(g\) is even. Finally, we get that for any \(n \in \{n \geq 3: n \equiv 2,6 \pmod{12}\) and \(n \neq 8\}\), there exists a \(\text{CQS}(g^n : s)\) for all \(g \equiv 0 \pmod{6}\), \(s \equiv 0 \pmod{2}\) and \(0 \leq s \leq g\).