In this paper we extend the definition of pseudograceful graphs given by Frucht [3] to all graphs \(G\) with vertex set \(V(G)\) and edge set \(E(G)\) such that
\(|V(G)| \leq |E(G)| + 1\) and we prove that if \(G\) is a pseudograceful graph, then \(G \cup K_{m,n}\).is pseudograceful
for \(m,n \geq 2\) and \((m,n) \neq (2,2)\) and is graceful for \(m,n \geq 2\). This enables us to obtain several new families of graceful and disconnected graphs.

A graph \(G\) is \(Z_m\)-well-covered if \(|I| \equiv |J| \pmod{m}\), for all \(I\), \(J\) maximal independent sets in \(V(G)\). A graph \(G\) is a \(1-Z_m\)-well-covered graph if \(G\) is \(Z_m\)-well-covered and \(G\setminus\{v\}\) is \(Z_m\)-well-covered, \(\forall v \in V(G)\). A graph \(G\) is strongly \(Z_m\)-well-covered if \(G\) is a \(Z_m\)-well-covered graph and \(G\setminus\{e\}\) is \(Z_m\)-well-covered, \(\forall e \in E(G)\). Here we prove some results about \(1-Z_m\)-well-covered and strongly \(Z_m\)-well-covered graphs.

There are two types of quadrangles in a projective plane, Fano quadrangles, and non-Fano quadrangles. The number of quadrangles in some small projective planes is counted according to type, and an interesting configuration in the Hughes plane is displayed.

Let \(S = T \sim (\cup\{A : A \in \mathcal{A}\})\), where \(T\) is a simply connected orthogonal polygon and \(\mathcal{A}\) is a collection of \(n\) pairwise disjoint open rectangular regions contained in \(T\). Point \(x\) belongs to the staircase kernel of \(S\), Ker \(S\), if and only if \(x\) belongs to Ker \(T\) and neither the horizontal nor the vertical line through \(x\) meets any \(A\) in \(\mathcal{A}\). This produces a Krasnosel’skii-type theorem for \(S\) in terms of \(n\). However, an example shows that, independent of \(n\), no general Krasnosel’skii number exists for \(S\).

We show that the secants of an arc of size near to \({\sqrt{2q}}\) cover almost half plane; also, a random union of \(log_2 q\) arcs of this size is such that its secants cover the plane.

Generalized Steiner triple systems, \(GS(2,3,n,g)\) are used to construct maximum constant weight codes over an alphabet of size \(g+1\) with distance \(3\) and weight \(3\) in which each codeword has length \(n\). The existence of \(GS(2,3,n,g)\) has been solved for \(g = 2,3,4,5,6,9\). The necessary conditions for the existence of a \(GS(2,3,n,g)\) are \((n-1)g \equiv 0 \pmod{2}\), \(n(n-1)g \equiv 0 \pmod{6}\), and \(n \geq g+2\). In this paper, the existence of a \(GS(2,3,7,g)\) for any given \(g \geq 7\) is investigated. It is proved that if there exists a \(GS(2,3,n,g)\) for all \(n\), \(g+2 \leq n \leq 9g+158\), satisfying the two congruences, then the necessary conditions are also sufficient. As an application it is proved that the necessary conditions for the existence of a \(GS(2,3,n,g)\) are also sufficient for \(g = 7,8\).

The Ramsey numbers \(r(C_5,G)\) are determined for all graphs \(G\) of order six.

For a graph \(G\), let \(Var(G)\) denote the variance of the degree sequence of \(G\), let \(sq(G)\) denote the sum of the squares of the degrees of \(G\), and let \(t(G)\) denote the number of triangles in \(G\) and in its complement. The parameters are related by:
\(Var(G) = \frac{sq(G)}{n} – d^2\)
where \(d\) is the average degree of \(G\), and
\(t(G) = \binom{n}{3} + \frac{sq(G)}{2} – {m(n-1)}\)
Let \(Var(n)\) denote the maximum possible value of \(Var(G)\) where \(G\) has \(n\) vertices, and let \(sq(n,m)\) and \(t(n,m)\) denote the maximum possible values of \(sq(G)\) and \(t(G)\), respectively, where \(G\) has \(n\) vertices and \(m\) edges. We present a polynomial time algorithm which generates all the graphs with \(n\) vertices and \(m\) edges having \(sq(G) = sq(n,m)\) and \(t(G) = t(n,m)\). This extends a result of Olpp which determined \(t(n,m)\). We also determine \(Var(n)\) precisely for every \(n\), and show that
\[ Var(n) = \frac{q(q-1)^2}{n}(1-\frac{q}{n}) =\frac{27}{256}n^2=O(n)\]
where \(q = [\frac{3n}{4}] \),(if \(n \equiv 2 \pmod 4\) the rounding is up ) thereby improving upon previous results.

This paper defines a new graph invariant by considering the set of connected induced subgraphs of a graph and defining a polynomial whose coefficients are determined by this partially ordered set of subgraphs. We compute the polynomial for a variety of graphs and also determine the effects on the polynomial of various graph operations.