Let \(g(n, k)\) be the maximum number of colors for the vertices of the cube graph \(Q_n\), such that each subcube \(Q_k\) contains all colors. Some exact values of \(g(n, k)\) are determined.

Let \( G \) be a connected graph and let \( U \) be a set of vertices in \( G \). A \({minimal \; U -tree}\) is a subtree \( T \) of \( G \) that contains \( U \) and has the property that every vertex of \( V(T) – U \) is a cut-vertex of \( \langle V(T) \rangle \). The \({monophonic\; interval}\) of \( U \) is the collection of all vertices of \( G \) that lie on some minimal \( U \)-tree. A set \( S \) of vertices of \( G \) is \( m_k \)-\({convex}\) if it contains the monophonic interval of every \( k \)-subset \( U \) of vertices of \( S \). Thus \( S \) is \( m_2 \)-convex if and only if it is \( m \)-convex.

In this paper, we consider three local convexity properties with respect to \( m_3 \)-convexity and characterize the graphs having either property.

Let \( G \) and \( H \) be graphs on \( n+2 \) vertices \( \{u_1, u_2, \ldots, u_n, x, y\} \) such that \( G – u_i \cong H – u_i \), for \( i = 1, 2, \ldots, n \). Recently, Ramachandran, Monikandan, and Balakumar have shown in a sequence of two papers that if \( n \geq 9 \), then \( |\varepsilon(H) – \varepsilon(G)| \leq 1 \). In this paper, we present a simpler proof of their theorem, using a counting lemma.

We answer in the affirmative a question posed by Al-Addasi and Al-Ezeh in 2008 on the existence of symmetric diametrical bipartite graphs of diameter 4. Bipartite symmetric diametrical graphs are called \( S \)-graphs by some authors, and diametrical graphs have also been studied by other authors using different terminology, such as self-centered unique eccentric point graphs. We include a brief survey of some of this literature and note that the existence question was also answered by Berman and Kotzig in a 1980 paper, along with a study of different isomorphism classes of these graphs using a \( (1,-1) \)-matrix representation which includes the well-known Hadamard matrices. Our presentation focuses on a neighborhood characterization of \( S \)-graphs, and we conclude our survey with a beautiful version of this characterization known to Janakiraman.

The achromatic number for a graph \( G = (V, E) \) is the largest integer \( m \) such that there is a partition of \( V \) into disjoint independent sets \( (V_1, \ldots, V_m) \) such that for each pair of distinct sets \( V_i, V_j \), \( V_i \cup V_j \) is not an independent set in \( G \). In this paper, we present an \( O(1) \)-approximation algorithm to determine the achromatic number of circulant graphs \( G(n; \pm\{1, 2\}) \) and \( G(n; \pm\{1, 2, 3\}) \).

Let \( G = (V, E) \) be a graph with vertex set \( V \) and edge set \( E \). Let \( diam(G) \) denote the diameter of \( G \) and \( d(u, v) \) denote the distance between the vertices \( u \) and \( v \) in \( G \). An antipodal labeling of \( G \) with diameter \( d \) is a function \( f \) that assigns to each vertex \( u \) a positive integer \( f(u) \), such that \( d(u, v) + |f(u) – f(v)| \geq d \), for all \( u, v \in V \). The span of an antipodal labeling \( f \) is \( \max\{|f(u) – f(v)| : u, v \in V(G)\} \). The antipodal number for \( G \), denoted by \( an(G) \), is the minimum span of all antipodal labelings of \( G \). Determining the antipodal number of a graph \( G \) is an NP-complete problem. In this paper, we determine the antipodal number of certain graphs with diameter equal to \( 3 \) and \( 4 \).

A radio labeling of a connected graph \( G \) is an injection \( f \) from the vertices of \( G \) to the natural numbers such that \( d(u, v) + |f(u) – f(v)| \geq 1 + \operatorname{diam}(G) \) for every pair of distinct vertices \( u \) and \( v \) of \( G \). The radio number of \( f \), denoted \( rn(f) \), is the maximum number assigned to any vertex of \( G \). The radio number of \( G \), denoted \( rn(G) \), is the minimum value of \( rn(f) \) taken over all labelings \( f \) of \( G \). In this paper, we determine bounds for the radio number of the hexagonal mesh.

In this paper, we introduce a finite graph using group characters and discuss the basic properties of the graph.

In this paper, a fuzzy inner product on a real vector space is introduced. The notion of fuzzy inner product is defined. Some of its properties are studied.

Let \( G = (V, E) \) be a simple graph. Let \( S \) be a subset of \( V(G) \). The toughness value of \( S \), denoted by \( T_S \), is defined as \( \frac{|S|}{\omega(G – S)} \), where \( \omega(G – S) \) denotes the number of components in \( G – S \). If \( S = V \), then \( \omega(G – S) \) is taken to be \( 1 \) and hence \( T_{V(G)} = |V(G)| \). A partition of \( V(G) \) into subsets \( V_1, V_2, \ldots, V_t \) such that \( T_{V_i} \), \( 1 \leq i \leq t \), is a constant is called an equi-toughness partitio of \( G \). The maximum cardinality of such a partition is called the equi-toughness partition number of \( G \) and is denoted by \( ET(G) \). The existence of \( ET \)-partition is guaranteed. In this paper, a study of this new parameter is initiated.