Let \( G \) be a simple graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). A labeling \( f : V(G) \to \mathbb{Z}_2 \) induces a partial edge labeling \( f^* : E(G) \to \mathbb{Z}_2 \) defined by \( f^*(uv) = f(u) \) if and only if \( f(u) = f(v) \). For \( i \in \mathbb{Z}_2 \), let \( V_f(i) = \{v \in V(G) : f(v) = i\} \) and \( e_f(i) = |\{e \in E(G) : f^*(e) = i\}| \). A labeling \( f \) is called a friendly labeling if \( |V_f(0) – V_f(1)| \leq 1 \). The \( BI(G) \), the balance index set of \( G \), is defined as \( \{|e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is friendly}\} \). This paper focuses on the balance index sets of generalized book and ear expansion graphs.

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 \({ R -total\; dominating\; function}\) 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 \({ R -total \;domination \;problem}\) 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 \( K_m \circ K_n \).