Let $M=\{v_1,v_2,\dots ,v_t\}$ be an ordered set of vertices in a graph $G$. Then $(d(u,v_1),d(u,v_2),\dots ,d(u,v_{\ell }))$ 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

$$\mathrm{cr}(K_{1,1,3,n})=4\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor +\lfloor \frac{3n}{2}\rfloor$$

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 $n$ ($n\ge 3$) has at least $\frac{n+1}{2}$ near-perfect matchings which miss different vertices of $G$.

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.

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.

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

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 $\pi =(v_1,v_2,\dots ,v_n,v_{n+1}=v_1)$, and let $d(\pi )=\sum _{i=1}^{n}d(v_i,v_{i+1})$. 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.

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

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 $n\in \{n\ge 3:n\equiv 2,6(\mathrm{mod}12)$ and $n\ne 8\}$, there exists a $\text{CQS}(g^n:s)$ for all $g\equiv 0(\mathrm{mod}6)$, $s\equiv 0(\mathrm{mod}2)$ and $0\le s\le g$.