Cyber insurance ratemaking (CIRM) is a procedure used to set rates (or prices) for cyber insurance products provided by insurance companies. Rate estimation is a critical issue for cyber insurance products. This problem arises because of the unavailability of actuarial data and the uncertainty of normative standards of cyber risk. Most cyber risk analyses do not consider the connection between Information Communication and Technology (ICT) sources. Recently, a cyber risk model was developed that considered the network structure. However, the analysis of this model remains limited to an unweighted network. To address this issue, we propose using a graph mining approach (GMA) to CIRM, which can be applied to obtain fair and competitive prices based on weighted network characteristics. This study differs from previous studies in that it adds the GMA to CIRM and uses communication models to explain the frequency of communications as weights in the network. We used the heterogeneous generalized susceptible-infectious-susceptible model to accommodate different infection rates. Our approach adds up to the existing method because it considers the communication frequency and GMA in CIRM. This approach results in heterogeneous premiums. Additionally, GMA can choose more active communications to reflect high communications contribution in the premiums or rates. This contribution is not found when the infection rates are the same. Based on our experimental results, it is apparent that this method can produce more reasonable and competitive prices than other methods. The prices obtained with GMA and communication factors are lower than those obtained without GMA and communication factors.

Let c be a k-coloring of a (not necessary connected) graph H. Let П= {C1, C2, · · ·, Ck } be the partition of V(H) induced by c, where Ci is partition class receiving color i. The color code c_П(v) of a vertex v ∈ H is the ordered k-tuple (d(v, C1), d(v, C2), · · ·, d(v, Ck )), where d(v, Ci) = min{d(v, x)|x ∈ Ci} for all i ∈ [1,k]. If all vertices of H have distinct color codes, then c is called a locating k-coloring of H. The locating-chromatic number of H, denoted by χ_L’ (H), is the smallest k such that H admits a locating- coloring with k colors. If there is no integer k satisfying the above conditions, then we say that χ’_L (H) = ∞. Note that if H is a connected graph, then χ_L’ (H) = χL(H). In this paper, we provide upper bounds for the locating-chromatic numbers of connectedgraphs obtained from disconnected graphs where each component contains a single dominant vertex.

The concept of a graph partition dimension was introduced by Chartrand et al. (1998). Let Π = {L1, L2, L3, · · ·, Lk } be a k-partition of V(G). The representation r(v|Π) of a vertex v with respect to Π is the vector (d(v, L1), d(v, L2), · · ·, d(v, Lk)). The partition Π is called a resolving partition of G if r(w|Π) ≠ r(v|Π) for all distinct w, v ∈ V(G). The partition dimension of a graph, denoted by pd(G), is the cardinality of a minimum resolving partition of G.

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph of order is a mapping of the set of edges of the graph to such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

In computer science, graphs are used in variety of applications directly or indirectly. Especially quantitative labeled graphs have played a vital role in computational linguistics, decision making software tools, coding theory and path determination in networks. For a graph G(V,E) with the vertex set V and the edge set E, a vertex k-labeling $\phi: V \rightarrow \{1,2,\dots, k\}$ is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f their $w_\phi(e) \ne w_\phi(f)$, where the weight of an edge $e=xy \in E(G)$ is $w_\phi(xy)=\phi(x)+\phi(y)$. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we determine the edge irregularity strengths of some chain graphs and the join of two graphs. We introduce a conjecture and open problems for researchers for further research.