In a communication network, several vulnerability measures are used to determine the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a graph as modeling a network, the edge-integrity of a graph is one \textbf{measure of graph vulnerability} and it is defined to be the minimum sum of the orders of a set of edges being removed and a largest remaining component. In this paper, the edge-integrity of graphs \(B_n\), \(H_n\), and \(E_p^t\), are calculated. Also, some results are given about edge-integrity of these graphs.

In this paper, it is shown that the necessary condition for the existence of a holey perfect Mendelsohn design (HPMD) with block size 5, type \(h^n\) and index \(\lambda\), namely, \(n \geq 5\) and \(\lambda n(n-1)h^2 \equiv 0 \pmod{5}\), is also sufficient for \(\lambda \geq 2\). The result guarantees the analogous existence result for group divisible designs (GDDs) of type \(h^n\) having block size 5 and index \(4\lambda\).

The computational complexity of the graph isomorphism problem is still unknown. We consider Cartesian products \(K_n \times K_m\) of two complete graphs \(K_n\) and \(K_m\). An acyclic orientation of such a Cartesian product is called a sequence graph because it has an application in production scheduling. It can be shown that the graph isomorphism problem on the class of these acyclic digraphs is solvable in polynomial time. We give numbers of non-isomorphic sequence graphs for small \(n\) and \(m\). The orientation on the cliques of a sequence graph can be interpreted as job orders and machine orders of a shop scheduling problem with a complete operation set.

Tenacity is a recently introduced parameter to measure vulnerability of networks and graphs. We characterize graphs having the maximum number of edges among all graphs with a given number of vertices and tenacity.

In this paper, we show that some graphs are circuit unique by applying a new tool, which is the character of the matching polynomial. Some properties of the character of the matching polynomial is also given.

The theory of hypergeometric functions is brought to bear on a problem—namely, that of obtaining a certain power series expansion involving the sine function that is inclusive of the Catalan sequence and which serves as a prelude to the calculation of other related series of similar type. A general formulation provides the particular result of interest as a special case, into which Catalan numbers are introduced as desired.

A splitting partition for a graph \(G = (V, E)\) is a partition of \(V\) into sets \(R\), \(B\), and \(U\) so that the subgraphs induced by \(V – R\) and \(V – B\) are isomorphic. The splitting number \(\mu(G)\) is the size of \(|R|\) for any splitting partition which maximizes \(|R|\). This paper determines \(\mu(G)\) for trees of maximum degree at most three and exactly one degree two vertex and for trees all of whose vertices have degree three or one.

A decomposition of a digraph is said to be bicyclic if it admits an automorphism consisting of exactly two disjoint cycles. Necessary and sufficient conditions are given for the existence of bicyclic decompositions of the complete digraph into each of the four orientations of a 4-cycle.

The integrity of a graph \(G\), \(I(G)\), is defined by \(I(G) = min_{S \subseteq V(G)}\{|S| + m(G – S)\}\) where \(m(G – S)\) is the maximum order of the components of \(G – S\). In general, the integrity of an \(r\)-regular graph is not known [8]. We answer the following question for special regular graphs. For any given two integers \(p\) and \(r\) such that \(\frac{pr}{2}\) is an integer, is there an \(r\)-regular graph, say \(G^*\), on \(p\) vertices having size \(q = \frac{pr}{2}\) such that
\[I(G(p,\frac{pr}{2})) \leq I(G^*)\]
for all \(p\) and \(r\)? The \emph{integrity graph} is denoted by \(IG(p,n)\). It is a graph with \(p\) vertices, integrity \(n\), and has the least number of edges denoted by \(q[p,n]\). We compute \(q[p,n]\) for some values of \(p,n\).