Given a graph \( G = (V, E) \), a set \( W \subseteq V \) is said to be a resolving set if for each pair of distinct vertices \( u, v \in V \), there is a vertex \( x \) in \( W \) such that \( d(u, x) \neq d(v, x) \). The resolving number of \( G \) is the minimum cardinality of all resolving sets. In this paper, a condition is imposed on resolving sets and a conditional resolving parameter is studied for grid-based networks.

Let \( G = (V, E) \) be a graph. A vertex labeling \( f: V \to \mathbb{Z}_2 \) induces an edge labeling \( f^*: E \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) + f(y) \) for each \( xy \in E \). For each \( i \in \mathbb{Z}_2 \), define \( v_f(i) = |f^{-1}(i)| \) and \( e_f(i) = |{f^*}^{-1}(i)| \). We call \( f \) friendly if \( |v_f(1) – v_f(0)| \leq 1 \). The full friendly index set of \( G \) is the set of all possible values of \( e_f(1) – e_f(0) \), where \( f \) is a friendly labeling. In this paper, we study the full friendly index set of the wheel \( W_n \), the tensor product of paths \( P_2 \) and \( P_n \), i.e., \( P_2 \otimes P_n \), and the double star \( D(m, n) \).

The detour order of a graph \( G \), denoted \( \tau(G) \), is the order of a longest path in \( G \). A partition \( (A, B) \) of \( V(G) \) such that \( \tau(\langle A \rangle) \leq a \) and \( \tau(\langle B \rangle) \leq b \) is called an \( (a, b) \)-partition of \( G \). A graph \( G \) is called \( \tau \)-\textit{partitionable} if \( G \) has an \( (a, b) \)-partition for every pair \( (a, b) \) of positive integers such that \( a + b = \tau(G) \).

The well-known Path Partition Conjecture states that every graph is \( \tau \)-partitionable. Motivated by the recent result of Dunbar and Frick [6] that if every \( 2 \)-connected graph is \( \tau \)-partitionable, then every graph is \( \tau \)-partitionable, we show that the Path Partition Conjecture is true for a large family of \( 2 \)-connected graphs with certain ear-decompositions. Also, we show that a family of \( 2 \)-edge-connected graphs with certain ear-decompositions is \( \tau \)-partitionable.

he problem of determining the collaboration graph of co-authors of Paul Erdos is a challenging task. Here we take up this problem for the case of Rolf Nevanlinna Prize Winners. Even though the number of prize winners as of date is 7, the collaboration graph has 20 vertices and 41 edges and possesses several interesting properties. In this paper, we have obtained this graph and determined standard graph parameters for the graph as well as its complement besides probing its structural properties. Several new results were obtained.

The topological descriptor Wiener index, named after the chemist Harold Wiener, is defined as half the sum of the distances between every pair of vertices of a graph. A lot of research has been devoted to finding the Wiener index by brute force method. In this paper, we compute the Wiener index of chemical structures such as sodium chloride and benzenoid without using a distance matrix.

A graph \( G(p, q) \) is said to be total edge bimagic with two common edge counts \( k_1 \) and \( k_2 \) if there exists a bijection \( f: V(G) \cup E(G) \to \{1, 2, \ldots, p + q\} \) such that for each edge \( uv \in E(G) \), \( f(u) + f(v) + f(uv) = k_1 \) or \( k_2 \).
A total edge-bimagic graph is called super edge-bimagic if \( f(V(G)) = \{1, 2, \ldots, p\} \). In this paper, we define new types of super edge-bimagic labeling and prove some interesting results related to super edge-bimagic labeling. Also, its relationship with cordial labeling is studied.

Trust is one of the most important means to improve the reliability of computing resources provided in a cloud environment and it plays an important role in commercial cloud environments. Trust is the estimation of the capability of a cloud resource in completing a task based on reputation, identity, behavior, and availability in the context of a distributed environment. It helps customers in the selection of appropriate resources in heterogeneous cloud infrastructure. The cloud computing depends on the following QoS parameters such as reliability, availability, scalability, security, and past behavior of the cloud resources.
This paper introduces a novel trust model to evaluate cloud resources of IaaS (Infrastructure as a Service) providers by means of Trust Resource Broker. The Trust Resource Broker selects trustworthy cloud resources based on the requirements of customers. The proposed trust model evaluates the trust value of the resources based on the identity as well as behavioral trust. The proposed model applies the QoS metrics suitable for cloud resources. The results of the experiments show that the proposed trust model selects the most reliable resources in a cloud environment.

A twofold 8-cycle system is an edge-disjoint decomposition of a twofold complete graph (which has two edges between every pair of vertices) into 8-cycles. The order of the complete graph is also called the order of the 8-cycle system. A twofold 2-perfect \(8\)-cycle system is a twofold \(8\)-cycle system such that the collection of distance \(2\) edges in each \(8\)-cycle also cover the complete graph, forming a (twofold) \(4\)-cycle system. Existence of \(2\)-perfect \(8\)-cycle systems for all admissible orders was shown in [1], although \(\lambda\)-fold existence for \(\lambda > 1\) has not been done.

In this paper, we impose an extra condition on the twofold \(2\)-perfect \(8\)-cycle system. We require that the two paths of length two between each pair of vertices, say \(x, a_{xy}, y\) and \(x, b_{xy}, y\), should be distinct, that is, with \(a_{xy} \neq b_{xy}\); thus they form a \(4\)-cycle \((x, a, y, b)\).

We completely solve the existence of such twofold \(2\)-perfect \(8\)-cycle systems with this “extra” property. All admissible orders congruent to \(0\) or \(1\) modulo \(8\) can be achieved, apart from order 8.

A graph \( G \) with \( k \) vertices is distance magic if the vertices can be labeled with numbers \( 1, 2, \ldots, k \) so that the sum of labels of the neighbors of each vertex is equal to the same constant \( \mu_0 \). We present a construction of distance magic graphs arising from arbitrary regular graphs based on an application of magic rectangles. We also solve a problem posed by Shafig, Ali, and Simanjuntak.

Given a graph \( G \), a function \( f : V(G) \to \{1, 2, \ldots, k\} \) is a \( k \)-ranking of \( G \) if \( f(u) = f(v) \) implies every \( u \)-\( v \) path contains a vertex \( w \) such that \( f(w) > f(u) \). A \( k \)-ranking is \emph{minimal} if the reduction of any label greater than \( 1 \) violates the described ranking property. The rank number of a graph, denoted \( \chi_r(G) \), is the minimum \( k \) such that \( G \) has a minimal \( k \)-ranking. The arank number of a graph, denoted \( \psi_r(G) \), is the maximum \( k \) such that \( G \) has a minimal \( k \)-ranking. It was asked by Laskar, Pillone, Eyabi, and Jacob if there is a family of graphs where minimal \( k \)-rankings exist for all \( \chi_r(G) \leq k \leq \psi_r(G) \). We give an affirmative answer showing that all intermediate minimal \( k \)-rankings exist for paths and cycles. We also give a characterization of all complete multipartite graphs which have this intermediate ranking property and which do not.