In this paper, we introduce the concept of similar graphs. Similar graphs arise in the design of fault-tolerant networks and in load balancing of the networks in case of node failures. Similar graphs model networks that not only remain connected but also allow a job to be shifted to other processors without re-executing the entire job. This dynamic load balancing capability ensures minimal interruption to the network in case of single or multiple node failures and increases overall efficiency. We define a graph to be \((m, n)\)-similar if each vertex is contained in a set of at least \(m\) vertices, each pair of which share at least \(n\) neighbors. Several well-known classes of \((2, 2)\)-similar graphs are characterized, for example, triangulated, comparability, and co-comparability. The problem of finding a minimum augmentation to obtain a \((2, 2)\)-similar graph is shown to be NP-Complete. A graph is called strongly \(m\)-similar if each vertex is contained in a set of at least \(m\) vertices with the property that they all share the same neighbors. The class of strongly \(m\)-similar graphs is completely characterized.

A star \(S_q\), with \(q\) edges, is a complete bipartite graph \(K_{1,q}\). Two figures of the complete graph \(K_n\) on a given set of \(k\) vertices are compatible if they are edge-disjoint, and a configuration is a set of pairwise compatible figures. In this paper, we take stars as our figures. A configuration \(C\) is said to be maximal if there is no figure (star) \(f \notin C\) such that \(\{f\} \cup C\) is also a configuration. The size of a configuration \(F\), denoted by \(|F|\), is the number of its figures. Let \(\text{Spec}(n, q)\) (or simply \(\text{Spec}(n)\)) denote the set of all sizes such that there exists a maximal configuration of stars with this size. In this paper, we completely determine \(\text{Spec}(n)\), the spectrum of maximal configurations of stars. As a special case, when \(n\) is an order of a star system, we obtain the spectrum of maximal partial star systems.

It is proved in this paper that for \(\lambda = 4\) and \(5\), the necessary conditions for the existence of a simple \(B(4, \lambda; v)\) are also sufficient. It is also proved that for \(\lambda = 4\) and \(5\), the necessary conditions for the existence of an indecomposable simple \(B(4, \lambda; v)\) are also sufficient, with the unique exception \((v, \lambda) = (7, 4)\) and \(10\) possible exceptions.

Let \(S\) and \(T\) be sets with \(|S| = m\) and \(|T| = n\). Let \(S_3, S_2\) and \(T_3, T_2\) be the sets of all \(3\)-subsets (\(2\)-subsets) of \(S\) and \(T\), respectively. Define \(Q((m, 2, 3), (n, 2, 3))\) as the smallest subset of \(S_2 \times T_2\) needed to cover all elements of \(S_3 \times T_3\). A more general version of this problem is initially defined, but the bulk of the investigation is devoted to studying this number. Its property as a lower bound for a planar crossing number is the reason for this focus.

Under some assumptions on the incidence matrices of symmetric designs, we prove a non-existence theorem for symmetric designs. The approach generalizes Wilbrink’s result on difference sets \([7]\).

In this paper, we derive some inequalities which the parameters of a two-symbol balanced array \(T\) (\(B\)-array) of strength four must satisfy for \(T\) to exist.