Figueroa-Centeno, Ichishima, and Muntaner-Batle [3, 4] proved some results on felicitous graphs and raised the following conjectures:

1. The one-point union of \( m \) copies of \( C_n \) is felicitous if and only if \( mn \equiv 2 \pmod{4} \).
2. \( mC_n \) is felicitous if and only if \( mn \not\equiv 2 \pmod{4} \).

In this paper, the conjectures are partially settled by proving the following results:

1. For any odd positive integers \( m \) and \( n \), the one-point union of \( m \) copies of \( C_n \) is felicitous if \( mn \equiv 1, 3 \).
2. For any positive integer \( m \), the one-point union of \( m \) copies of \( C_4 \) is felicitous.
3. For any two odd positive integers \( m \) and \( n \), \( mC_n \) is felicitous if \( mn \equiv 1, 3 \pmod{4} \).
4. For any positive integer \( m \), \( mC_4 \) is felicitous.

In this paper, we characterize the graphs \( G \) and \( H \) for which the Cartesian product \( G \Box H \) is a divisor graph. We show that divisor graphs form a proper subclass of perfect graphs. Additionally, we prove that cycle permutation graphs of order at least 8 are divisor graphs if and only if they are perfect. Some results concerning amalgamation operations for obtaining new divisor graphs from old ones are presented. We view block graphs as vertex amalgams.

This note will complete the computation of all Ramsey numbers \( r(G, H) \) for graphs \( G \) of order at most five and disconnected graphs \( H \) of order six.

For a graph \( G \) and a real number \( \alpha \neq 0 \), the graph invariant \( s_\alpha^+(G) \) is the sum of the \( \alpha \)th power of the non-zero signless Laplacian eigenvalues of \( G \). In this paper, several lower and upper bounds for \( s_\alpha^+(G) \) with \( \alpha \neq 0, 1 \) are obtained. Applying these results, we also derive some bounds for the incidence energy of graphs, which generalize and improve on some known results.

Any \( H \)-free graph \( G \) is called \( H \)-saturated if the addition of any edge \( e \notin E(G) \) results in \( H \) as a subgraph of \( G \). The minimum size of an \( H \)-saturated graph on \( n \) vertices is denoted by \( sat(n, H) \). The edge spectrum for the family of graphs with property \( P \) is the set of all sizes of graphs with property \( P \). In this paper, we find the edge spectrum of \( K_4 \)-saturated graphs. We also show that if \( G \) is a \( K_4 \)-saturated graph, then either \( G \cong K_{1,1,n-2} \) or \( \delta(G) \geq 3 \), and we detail the exact structure of a \( K_4 \)-saturated graph with \( \kappa(G) = 2 \) and \( \kappa(G) = 3 \).

The Hosoya index of a graph is defined as the summation of the coefficients of the matching polynomial of a graph. In this paper, we give an explicit expression of the Hosoya index for the graphs \( C(n, v_1v_i) \), \( Q(n, v_1v_s) \), and \( D(s, t) \), and also characterize the extremal graphs with respect to the upper and lower bounds of the Hosoya index of these graphs. In particular, we provide the Hosoya index order for the graphs \( C(n, v_1v_i) \) and \( Q(n, v_1v_s) \), respectively.

Let \( \mathcal{P} = \{I, I_1+d, I_1+2d, \ldots, I_1+(\ell-1)d\} \), where \( \ell, d, I_1 \) are fixed integers and \( \ell, d > 0 \). Suppose that \( G = (V, E) \) is a graph and \( R \) is a labeling function which assigns an integer \( R(v) \) to each \( v \in V \). An \emph{\( R \)-total dominating function} of \( G \) is a function \( f: V \to \mathcal{P} \) such that

\[
\sum_{u \in N_G(v)} f(u) \geq R(v)
\]

for all vertices \( v \in V \), where \( N_G(v) = \{u \mid (u, v) \in E\} \). The \emph{\( R \)-total domination problem} is to find an \( R \)-total dominating function \( f \) of \( G \) such that

\[
\sum_{v \in V} f(v)
\]

is minimized. In this paper, we present a linear-time algorithm to solve the \( R \)-total domination problem on convex bipartite graphs. Our algorithm gives a unified approach to the \( k \)-total, signed total, and minus total domination problems for convex bipartite graphs.

The Laplacian eigenvalues of linear phenylenes \( PH_n \) are partially determined, and a simple closed-form formula for the Kirchhoff index of \( PH_n \) is derived in terms of the index \( n \).

The notion of equitable coloring was introduced by Meyer in 1973. This paper presents exact values of the equitable chromatic number of three corona graphs, which include the complete graph and its complement \( K_m \circ \overline{K_n} \), the star graph and its complement \( K_{1,m} \circ \overline{K_{1,n}} \), and the complete graph and complete graph \( K_m \circ K_n \).

A construction of graphs, codes, and designs acted on by simple primitive groups described in [9, 10] is used to find some self-orthogonal, irreducible, and indecomposable codes acted on by one of the simple Janko groups, \( J_1 \) or \( J_2 \). In particular, most of the irreducible modules over the fields \( \mathbb{F}_p \) for \( p \in \{2, 3, 5, 7, 11, 19\} \) for \( J_1 \), and \( p \in \{2, 3, 5, 7\} \) for \( J_2 \), can be represented in this way as linear codes invariant under the groups.

Let \( G = (V_1, V_2; E) \) be a bipartite graph with \( |V_1| = |V_2| = 2k \), where \( k \) is a positive integer. It is proved that if \( d(x) + d(y) \geq 3k \) for every pair of nonadjacent vertices \( x \in V_1 \), \( y \in V_2 \), then \( G \) contains \( k \) independent quadrilaterals.

A set \( S \) of vertices of a graph \( G \) is geodetic if every vertex in \( V(G) \setminus S \) is contained in a shortest path between two vertices of \( S \). The geodetic number \( g(G) \) is the minimum cardinality of a geodetic set of \( G \). The geodomatic number \( d_g(G) \) of a graph \( G \) is the maximum number of elements in a partition of \( V(G) \) into geodetic sets.

In this paper, we determine \( d_g(G) \) for some family of graphs, and we present different bounds on \( d_g(G) \). In particular, we prove the following Nordhaus-Gaddum inequality, where \( \overline{G} \) is the complement of the graph \( G \). If \( G \) is a graph of order \( n \geq 2 \), then
\[
d_g(G) + d_g(\overline{G}) \leq n
\]
with equality if and only if \( n = 2 \).

For given finite simple graphs \( F \) and \( G \), the Ramsey number \( R(F, G) \) is the minimum positive integer \( n \) such that for every graph \( H \) of order \( n \), either \( H \) contains \( F \) or the complement of \( H \) contains \( G \). In this note, with the help of computer, we get that
\[
R(C_5, W_6) = 13, \quad R(C_5, W_7) = 15, \quad R(C_5, W_8) = 17,
\]
\[
R(C_6, W_6) = 11, \quad R(C_6, W_7) = 16, \quad R(C_6, W_8) = 13,
\]
\[
R(C_7, W_6) = 13 \quad \text{and} \quad R(C_7, W_8) = 17.
\]

A \((p,q)\)-graph is said to be a permutation graph if there exists a bijection function \( f: V(G) \to \{1, 2, \ldots, p\} \) such that the induced edge function \( h_f: E(G) \to \mathbb{N} \) is defined as follows:
\[
h_f(x_i, x_j) =
\begin{cases}
{}^{f(x_i)}P_{f(x_j)}, & \text{if } f(x_j) < f(x_i); \\ {}^{f(x_j)}P_{f(x_i)}, & \text{if } f(x_i) < f(x_j). \end{cases} \] In this paper, we investigate the permutation labelings of wheel-related graphs.

Determining whether or not a graph has an efficient dominating set (equivalently, a perfect code) is an NP-complete problem. Here we present a polynomial time algorithm to decide if a given simplicial graph has an efficient dominating set. However, the efficient domination number decision problem is NP-complete for simplicial graphs.

The purpose of this note is to give two binomial sums with generalized Fibonacci sequences. These results generalize two binomial sums by Kilic and Ionascu in The Fibonacci Quarterly, 48.2(2010), 161-167.

Let \( G \) be a connected graph of size at least 2 and \( c: E(G) \to \{0, 1, \ldots, k-1\} \) an edge coloring (or labeling) of \( G \) using \( k \) colors (where adjacent edges may be assigned the same color). For each vertex \( v \) of \( G \), the color code of \( v \) with respect to \( c \) is the \( k \)-tuple \( \text{code}(v) = (a_0, a_1, \ldots, a_{k-1}) \), where \( a_i \) is the number of edges incident with \( v \) that are labeled \( i \) (for \( 0 \leq i \leq k-1 \)). The labeling \( c \) is called a detectable labeling if distinct vertices in \( G \) have distinct color codes. The value \( \text{val}(c) \) of a detectable labeling \( c \) of a graph \( G \) is the sum of the colors assigned to the edges in \( G \). The total detection number \( \text{td}(G) \) of \( G \) is defined by \( \text{td}(G) = \min\{\text{val}(c)\} \), where the minimum is taken over all detectable labelings \( c \) of \( G \). Thus, if \( G \) is a connected graph of size \( m \geq 2 \), then \( 1 \leq \text{td}(G) \leq \binom{m}{2} \). We present characterizations of all connected graphs \( G \) of size \( m \geq 2 \) for which \( \text{td}(G) \in \{1, \binom{m}{2}\} \). The total detection numbers of complete graphs and cycles are also investigated.

In this paper we prove that every planar graph without \(5\)- and \(8\)-cycles and without adjacent triangles is \(3\)-colorable.

A new construction of authentication codes with arbitration using singular pseudo-symplectic geometry on finite fields is given. Some parameters and the probabilities of success for different types of deceptions are computed.

Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. By \( h(G,x) \) and \( P(G,\lambda) \) we denote the adjoint polynomial and the chromatic polynomial of graph \( G \), respectively. A new invariant of graph \( G \), which is the fifth character \( R_5(G) \), is given in this paper. Using this invariant and the properties of the adjoint polynomials, we firstly and completely determine the adjoint equivalence class of the graph \( \zeta_n^1 \). According to the relations between \( h(G,x) \) and \( P(G,\lambda) \), we also simultaneously determine the chromatic equivalence class of \( \overline{\zeta_n^1} \).

In this paper, we discuss the properties of a class of generalized harmonic numbers \( H_{n,r} \). Using Riordan arrays and generating functions, we establish some identities involving \( H_{n,r} \). Furthermore, we investigate certain sums related to harmonic polynomials \( H_n(z) \). In particular, using the Riordan array method, we explore interesting relationships between these polynomials, the generating Stirling polynomials, the Bernoulli polynomials, and the Cauchy polynomials. Finally, we obtain the asymptotic expansion of certain sums involving \( H_{n,r} \).

We prove that \( F_v(3,5;6) = 16 \), which solves the smallest open case of vertex Folkman numbers of the form \( F_v(3, k; k+1) \). The proof uses computer algorithms.

A family \( \mathcal{G} \) of connected graphs is a family with constant metric dimension if \( \dim(G) \) is finite and does not depend upon the choice of \( G \) in \( \mathcal{G} \). The metric dimension of some classes of plane graphs has been determined in references [3], [4], [5], [12], [14], and [18], while the metric dimension of some families of convex polytopes has been studied in references [8], [9], [10], and [11]. The following open problem was raised in reference [11].

Open Problem [11]: Let \( G \) be the graph of a convex polytope which is obtained by joining the graph of two different convex polytopes \( G_1 \) and \( G_2 \) (such that the outer cycle of \( G_1 \) is the inner cycle of \( G_2 \)) both having constant metric dimension. Is it the case that \( G \) will always have constant metric dimension?

In this paper, we extend this study to an infinite class of convex polytopes obtained as a combination of the graph of an antiprism \( A_n \) [1] and the graph of convex polytope \( Q_n \) [2], such that the outer cycle of \( A_n \) is the inner cycle of \( Q_n \). It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension. Note that the problem of determining whether \( \dim(G) < k \) is an NP-complete problem [7].