Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A = \{0, 1\} \). A labeling \( f: V(G) \to A \) induces an edge partial labeling \( f^*: E(G) \to A \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E(G) \). For each \( i \in A \), let \(v_f(i) = |\{v \in V(G) : f(v) = i\}|\) and \(e_f(i) = |\{e \in E(G) : f^*(e) = i\}|.\)The balance index set of \( G \), denoted \( BI(G) \), is defined as \(\{|e_f(0) – e_f(1)|: |v_f(0) – v_f(1)| \leq 1\}.\)In this paper, exact values of the balance index sets of five new families of one-point union of graphs are obtained, many of them, but not all, form arithmetic progressions.

For any \( h \in \mathbb{Z} \), a graph \( G = (V, E) \) is said to be \( h \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_h – \{0\} \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_h \), defined by
\[
l^+(v) = \sum_{uv \in E(G)} l(uv)\]
is a constant map. For a given graph \( G \), the set of all \( h \in \mathbb{Z}_+ \) for which \( G \) is \( h \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). In this paper, we will determine the integer-magic spectra of trees of diameter five.

A graceful labeling of a directed graph \( D \) with \( e \) edges is a one-to-one map \( \theta: V(D) \to \{0, 1, \dots, e\} \) such that \( \theta(y) – \theta(x) \mod (e + 1) \) is distinct for each \( (x, y) \in E(D) \). This paper summarizes previously known results on graceful directed graphs and presents some new results on directed paths, stars, wheels, and umbrellas.

For an integer \( l > 1 \), the \( l \)-edge-connectivity of a graph \( G \) with \( |V(G)| \geq l \), denoted by \( \lambda_l(G) \), is the smallest number of edges whose removal results in a graph with \( l \) components. In this paper, we study lower bounds of \( \lambda_l(G) \) and optimal graphs that reach the lower bounds. Former results by Boesch and Chen are extended.

We also present in this paper an optimal model of interconnection network \( G \) with a given \( \lambda_l(G) \) such that \( \lambda_2(G) \) is maximized while \( |E(G)| \) is minimized.

Given an abelian group \( A \), a graph \( G = (V, E) \) is said to have a distance two magic labeling in \( A \) if there exists a labeling \( l: E(G) \to A – \{0\} \) such that the induced vertex labeling \( l^*: V(G) \to A \) defined by

\[l^*(v) = \sum_{c \in E(v)} l(e)\]

is a constant map, where \( E(v) = \{e \in E(G) : d(v,e) < 2\} \). The set of all \( h \in \mathbb{Z}_+ \), for which \( G \) has a distance two magic labeling in \( \mathbb{Z}_h \), is called the distance two magic spectrum of \( G \) and is denoted by \( \Delta M(G) \). In this paper, the distance two magic spectra of certain classes of graphs will be determined.

In this paper, we derive some necessary existence conditions for a bi-level balanced array (B-array) with strength \( t = 5 \). We then describe how these existence conditions can be used to obtain an upper bound on the number of constraints of these arrays, and give some illustrative examples to this effect.

Let \( G = (V, E) \) be a graph with a vertex labeling \( f: V \to \mathbb{Z}_2 \) that induces an edge labeling \( f^*: E \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) + f(y) \). For each \( i \in \mathbb{Z}_2 \), let \(
v_f(i) = \text{card}\{v \in V: f(v) = i\}\) and \(e_f(i) = \text{card}\{e \in E: f^*(e) = i\}.\) A labeling \( f \) of a graph \( G \) is said to be friendly if \(\lvert v_f(0) – v_f(1) \rvert \leq 1.\) The friendly index set of \( G \) is defined as \(\{\lvert e_f(1) – e_f(0) \rvert : \text{the vertex labeling } f \text{ is friendly}\}.\)
In this paper, we determine the friendly index sets of generalized books.

Given 2 triangles in a plane over a field \( F \) which are in perspective from a vertex \( V \), the resulting Desargues line or axis \( l \) may or may not be on \( V \). To avoid degenerate cases, we assume that the union of the vertices of the 2 triangles is a set of six points with no three collinear. Our work then provides a detailed analysis of situations when \( V \) is on \( l \) for any \( F \), finite or infinite.

We give constructive and combinatorial proofs to decide why certain families of slightly irregular graphs have no planar representation and why certain families have such planar representations. Several non-existence results for infinite families as well as for specific graphs are given. For example, the nonexistence of the graphs with \( n = 11 \) and degree sequence \( (5, 5, 5, \ldots, 4) \) and \( n = 13 \) and degree sequence \( (6, 5, 5, \ldots, 5) \) are shown.

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). Let \( A = \{0, 1\} \). A labeling \( f: V(G) \to A \) induces a partial edge labeling \( f^*: E(G) \to A \) defined by \(f^*(xy) = f(x) \quad \text{if and only if } f(x) = f(y),\) for each edge \( xy \in E(G) \). For \( i \in A \), let \(
v_f(i) = \text{card}\{v \in V(G) : f(v) = i\}\) and \(e_{f^*}(i) = \text{card}\{e \in E(G) : f^*(e) = i\}.\) A labeling \( f \) of a graph \( G \) is said to be friendly if \(\lvert v_f(0) – v_f(1) \rvert \leq 1.\)If \(\lvert e_{f^*}(0) – e_{f^*}(1) \rvert \leq 1,\) then \( G \) is said to be \(\textbf{balanced}\). The balancedness of the Cartesian product and composition of graphs is studied in [19]. We provide some new families of balanced graphs using other constructions.