An \((2,1)\) \text{coloring} of a graph \( G = (V,E) \) is a vertex coloring \( f : V(G) \rightarrow \{0,1,2,\ldots,k\} \) such that \( |f(u) – f(v)| \geq 2 \) for all \( uv \in E(G) \) and \( |f(u) – f(v)| \geq 1 \) if \( d(u,v) = 2 \). The \emph{span} \( \lambda(G) \) is the smallest \( k \) for which \( G \) has an \( L(2,1) \) \text{coloring}. A \text{span coloring} is an \( L(2,1) \) coloring whose greatest color is \( \lambda(G) \). An \( L(2,1) \)-\text{coloring} \( f \) is a full-coloring if \( f : V(G) \rightarrow \{0,1,2,\ldots,\lambda(G)\} \) is onto and \( f \) is an irreducible no-hole coloring (inh-coloring) if \( f : V(G) \rightarrow \{0,1,2,\ldots,k\} \) is onto for some \( k \) and there does not exist an \( L(2,1) \)-\text{coloring} \( g \) such that \( g(u) < f(u) \) for all \( u \in V(G) \) and \( g(v) < f(v) \) for some \( v \in V(G) \). The Assignment sum of \( f \) on \( G \) is the sum of all the labels assigned to the vertices of \( G \) by the \( L(2,1) \) \text{coloring} \( f \). The \emph{Sum coloring number} of \( G \), introduced in this paper, \( \sum(G) \), is the minimum assignment sum over all the possible \( L(2,1) \) colorings of \( G \). \( f \) is a \text{Sum coloring} on \( G \) if its assignment sum equals the \emph{Sum coloring number}. In this paper, we investigate the \emph{Sum coloring numbers} of certain classes of graphs. It is shown that \( \sum(P_n) = 2(n – 1) \) and \( \sum(C_n) = 2n \) for all \( n \). We also give an exact value for the Sum coloring number of a star and conjecture a bound for the Sum coloring number of an arbitrary graph \( G \) with span \( \lambda(G) \).

Let \( U(n, f) \) denote the graph with vertex set the set of unlabeled graphs of order \( n \) that have no vertex of degree greater than \( f \). Two vertices \( H \) and \( G \) of \( U(n, f) \) are adjacent if and only if \( H \) and \( G \) differ (up to isomorphism) by exactly one edge. The problem of determining the values of \( n \) and \( f \) for which \( U(n, f) \) contains a Hamilton path is investigated. There are only a few known non-trivial cases for which a Hamilton path exists. Specifically, these are \( U(5, 3) \), \( U(6, 3) \), and \( U(7, 3) \). On the other hand, there are many cases for which it is shown that no Hamilton path exists. The complete solution of this problem is unresolved.

We study nega-cyclic \(\pm 1\) matrices. We obtain preliminary results which are then used to decrease the search space. We find that there are \( 2 \), \( 4 \), \( 9 \), \( 23 \), \( 63 \), and \( 187 \) ip-equivalence classes for lengths \( 3 \), \( 5 \), \( 7 \), \( 9 \), \( 11 \), and \( 13 \) respectively. The matrices we find are used in a variant given here of the Goethals-Seidel array to form Hadamard matrices, the aim being to later check them for suitability for CDMA schemes.