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 \( K_m \circ K_n \).

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.