It is shown that the basis graphs of every family of circulants are characterized by their matching polynomials. Explicit formulas are also given for their matching polynomials. From these results, the analogous formulas for the chromatic polynomials of the complements of the basis graphs, are obtained. It is shown that a basis graph of a family of circulants is chromatically unique if and only if it is connected. Also, some interesting results of a computer investigation are discussed and conjectures are made.

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 \([7]\).