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On a graph G=(V,E), an Extended Roman Domination function is f: V → {0,1,2,3} satisfying the criteria that (i) every vertex u for which f(u) is either 0 or 1 is adjacent to at least one vertex v for which f(v) = 3, and(ii) if u and v are two adjacent vertices and if f(u) = 0 then f(v) ≠ 0. The sum of values assigned to all vertices determines the weight of an Extended Roman Domination function. The Extended Roman Domination number of G, indicated by γ erd is the minimal weight of an Extended Roman Domination function on a graph G. The ‘extended Roman domination’ of ‘sunlet, pan, wheel, gear, helm, and web graph’ is investigated in this research.

The degree distance of a graph is a graph invariant that is more sensitive than the Wiener index. In this paper, we calculate the vertex degree distances of one vertex union of the cycle and the star, and the degree distance of one vertex union of the cycle and the star. These results lay a foundation for further study on the extremal values of the vertex degree distances, and the distribution of the vertices with the extremal values in one vertex union of the cycle and the star.

The Gutman index is a variant of the Wiener index, which related to the degree and the distance. In this paper, we study the vertex Gutman indices of a special kind of graphs, one vertex union of two cycles. And the maximum and minimum of vertex Gutman indices are studied. The results show that the vertex Gutman indices has the maximum at the center and has minimum at some vertices nearest the center.

Given a simple graph G, a subset S ⊆ V ( G ) is an independent [1, 2]-set if no two vertices in S are adjacent and for every vertex υ ϵ V ( G )\\ S , 1 ≤ | N (υ) ∩ S | ≤ 2, that is, every vertex υ ϵ V ( G )\\ S is adjacent to at least one but not more than two vertices in S. This paper investigates the existence of independent [1, 2]-sets of hypercubes. We show that for some positive integer k, if n = 2 k − 1, then hypercubes Q n and Q n +1 have an independent [1, 2]-set. Furthermore, for 1 ≤ n ≤ 4, we find bounds for the minimum and maximum cardinality of an independent [1, 2]-set of hypercube Q n , while for n = 5, 6, we get the maximum of cardinality of an independent [1, 2]-set of hypercube Q n .

The injective chromatic number χ i (G) of a graph G is the minimum number of colors needed to color the vertices of G such that different vertices with the same neighbor are colored differently. In this here, we get some exact values of the injective chromatic number of the edge corona product of some graphs.

Objective The aim of the present pilot study was to assess the effectiveness of the platelet-rich fibrin (PRF) apical barrier for the placement of MTA for the treatment of teeth with periapical lesions and open apices. Methods A total of thirty teeth on twenty-eight patients with open apices and periapical periodontitis were enrolled and divided into two groups in the present pilot study. In the PRF group (fourteen teeth in thirteen patients), nonsurgical endodontic treatment was performed using PRF as an apical matrix, after which the apical plug of the MTA was created. For the non-PRF group (fourteen teeth in fourteen patients), nonsurgical endodontic therapy was performed using only the MTA for an apical plug with no further periapical intervention. Clinical findings and periapical digital radiographs were used for evaluating the healing progress after periodic follow-ups of 1, 3, 6, and 9 months. The horizontal dimension of the periapical lesion was gauged, and the changes in the dimensions were recorded each time. The Friedman test, Dunn-Bonferroni post hoc correction, and Mann-Whitney U test were used for statistical analysis, with P  < 0.05 serving as the threshold for determining statistical significance. Results All patients in both groups in the present pilot study had no clinical symptoms after 1 month, with a significant reduction in the periapical lesion after periodic appointments. The lesion width of the PRF group was significantly smaller than that of the non-PRF group in the sixth and ninth month after treatment. Conclusions PRF is a promising apical barrier matrix when combined with MTA for the treatment of teeth with open apices and periapical periodontitis. Small number of study subjects and the short time of follow-up period limit the generalizability of these results. Trial registration TCTR, TCTR20221109006. Registered 09 November 2022 - Retrospectively registered, https://www.thaiclinicaltrials.org/show/TCTR20221109006 .

Achieving an apical seal is critical for apexification treatment of nonvital immature teeth. While this is commonly accomplished using biocompatible mineral trioxide aggregate (MTA), its limitations, such as prolonged setting time, discoloration, and challenging handling, have driven the search for alternative materials. This study aimed to compare the clinical and radiographic success of bioceramic putty Well-Root PT apical plug compared to MTA in the treatment of nonvital immature permanent incisors. Fifty immature nonvital maxillary permanent central incisors in thirty-eight children aged 8–11 years were randomly divided into two groups (25 teeth/group). Group I received MTA apical plugs, and Group II was treated with Well-Root PT apical plugs. Both groups were recalled at 6 and 12 months for clinical and radiographic evaluations. Statistical analysis was done for the gathered data. Both groups showed improved clinical signs and symptoms during all follow-up periods with no statistically significant difference. Regarding the periapical radiolucency (PAR) area, at twelve months, the mean PAR area in the Well-Root PT group was (0.14 ± 0.08) compared to (2.3 ± 0.9) in the MTA group, with highly statistically significant differences (p < 0.001). The mean periapical bone radiodensity in the Well-Root PT group was (178.2 ± 5.4) compared to (164.8 ± 9.4) in the MTA group at twelve-month follow-up, with highly statistically significant differences(p < 0.001). Well-Root PT, with its reduced technical sensitivity, demonstrates satisfactory clinical and radiographic success as an apical plug for nonvital immature permanent incisors compared to MTA.

Let G be a nontrivial connected graph and let c : V ( G ) → ℕ be a vertex coloring of G , where adjacent vertices may have the same color. For a vertex υ of G , the color sum σ ( υ ) of υ is the sum of the colors of the vertices adjacent to υ . The coloring c is said to be a sigma coloring of G if σ ( u ) ≠ σ ( υ ) whenever u and υ are adjacent vertices in G . The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic number of G and is denoted by σ ( G ). In this study, we investigate sigma coloring in relation to a unary graph operation called middle graph. We will show that the sigma chromatic number of the middle graph of any path, cycle, sunlet graph, tadpole graph, ladder graph, or triangular snake graph is 2 except for some small cases. We also determine the sigma chromatic number of the middle graph of stars.

Minimum dominating set is a basic graph problem. Most existing solving algorithms are designed for static graphs. In this paper, an incremental update algorithm is proposed to solve the minimal dominating set of a dynamic graph. This algorithm can quickly update the MDS when the structure of graph changes, and not need to recalculate based on the entire graph. By analyzing the characteristics of the four structural changes (adding vertices, deleting vertices, adding edges, and deleting edges) in the graph, a local update strategy for the minimal dominating set is designed, and a reduction rule for the minimal dominating set is proposed. This not only effectively reduces the computational complexity, but also enables the algorithm results to approach the minimum dominating set. Compared with traditional static algorithms, our algorithm has higher efficiency and accuracy in calculating the minimal dominating set of dynamic graphs.

In this paper, we study the strong zero-divisor graph of a p.q.-Baer $*$-ring and establish conditions, based on the smallest central projection in the lattice of central projections, under which the graph contains a cut vertex. We prove that the set of cut vertices forms a complete subgraph. Furthermore, we show that the complement of this graph is connected if and only if the $*$-ring contains at least six central projections. The diameter and girth of the complement are determined, and we characterize p.q.-Baer $*$-rings whose strong zero-divisor graph is complemented.