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Definition of graph is set pair (𔑉(ð”º),ð”¸(ð”º)) where 𔑉(ð”º) is vertex set and ð”¸(ð”º) is edge set. A maping 𔼠: 𔑉(ð”º)→{1,2, ... ,𔑘} as label function and weight function 𔑤 : 𔑉(ð”º)→𔑠is desined as 𔑤(𔑢)=Σ𔑣∈ð”‘(𔑢)𔑙(𔑣). The function 𔑤 is called local irregularity vertex coloring if: (i) 𔑜𔑔𔑡(𔑙)=𔑚𔑖𔑛 (𔑚𔑎𔑘𔑠(𔑙𔑖) ;𔑙𔑖 𔑖𔑠𔑙𔑎ð”‘𔑒𔑙 𔑓𔑢𔑛ð”‘𔑡𔑖𔑜𔑛) and (ii) for every 𔑢𔑣 ∈ ð”¸(ð”º),𔑤(𔑢) ≠𔑤(𔑣). The chromatic number of local irregularity vertex coloring denoted by 𔜒𔑙𔑖𔑠(ð”º) is defined as 𔜒𔑙𔑖𔑠(ð”º)=𔑚𔑖𔑛{|𔑤(𔑉(ð”º))|;𔑤 𔑖𔑠𔑙𔑜ð”‘𔑎𔑙 𔑖𔑟𔑟𔑒𔑔𔑢𔑙𔑎𔑟𔑖𔑡𔑦 𔑣𔑒𔑟𔑡𔑒𔑥 ð”‘𔑜𔑙𔑜𔑟𔑖𔑛𔑔}. The method used in this paper is pattern recognition and axiomatic deductive method. In this paper, we learn local irregularity vertex coloring of vertex amalgamation of path graph and determine the chromatic number on local irregularity vertex coloring of vertex amalgamation of path graph. This paper use vertex amalgamation of path graph (𔑎𔑚𔑎𔑙(𔑃𔑛 ,𔑣,𔑚)). The result of this study are expected to be used as basic studies and science development as well as applications related to local irregularity vertex coloring of vertex amalgamation of path graph.

The local metric dimension is one of many topics in graph theory with several applications. One of its applications is a new model for assigning codes to customers in delivery services. Let $ G $ be a connected graph and $ V(G) $ be a vertex set of $ G $. For an ordered set $ W = \{ x_1, x_2, \ldots, x_k\} \subseteq V(G) $, the representation of a vertex $ x $ with respect to $ W $ is $ r_G(x|W) = \{(d(x, x_1), d(x, x_2), \ldots, d(x, x_k) \} $. The set $ W $ is said to be a local metric set of $ G $ if $ r(x|W)\neq r(y|W) $ for every pair of adjacent vertices $ x $ and $ y $ in $ G $. The eccentricity of a vertex $ x $ is the maximum distance between $ x $ and all other vertices in $ G $. Among all vertices in $ G $, the smallest eccentricity is called the radius of $ G $ and a vertex whose eccentricity equals the radius is called a central vertex of $ G $. In this paper, we developed a new concept, so-called the central local metric dimension by combining the concept of local metric dimension with the central vertex of a graph. The set $ W $ is a central local metric set if $ W $ is a local metric set and contains all central vertices of $ G $. The minimum cardinality of a central local metric set is called a central local metric dimension of $ G $. In the main result, we introduce the definition of the central local metric dimension of a graph and some properties, then construct the central local metric dimensions for trees and establish results for the grid graph.

The central local metric dimension is a new variation of local metric dimension that introduced in 2023. The central local metric dimension is a new concept that enriches research studies in graph theory, especially in the field of metric dimension. This concept combines the concept of local metric dimension by involving central points in the local metric set so that the existence of central points can strengthen the position of the local metric set in distinguishing every two neighboring points in a graph. The methodology of this research is study literature and observation. We find the central vertex of each graph and also find the local metric set of its graph, then we applied it to the related theorem to find the lower bound of central local metric dimension. Let 𝐺 be a connected graph with order 𝑛 and vertex set is 𝑉(𝐺). A subset 𝑊={𝑥1,𝑥2,…,𝑥𝑘}⊆𝑉(𝐺) is a local metric set of graph 𝐺 if the metric code of every two adjacent vertices 𝑢,𝑣 in 𝐺 are 𝑟(𝑢|𝑊)≠𝑟(𝑣|𝑊), where 𝑟(𝑢|𝑊)=(𝑑(𝑢,𝑥1),𝑑(𝑢,𝑥2),…,𝑑(𝑢,𝑥𝑘)) and 𝑟(𝑣|𝑊)=(𝑑(𝑣,𝑥1),𝑑(𝑣,𝑥2),…,𝑑(𝑣,𝑥𝑘)). A vertex 𝑥∈𝑉(𝐺) is a central vertex in 𝐺 if 𝑥 have the the shorthest distance to the another all vertices in 𝐺. If 𝑊 consist of all central vertices in 𝐺, then 𝑊 is called a central local metric set of 𝐺. The minimal cardinality of 𝑊 is called a central basis local set of 𝐺 and its cardinality is called central local metric dimension of 𝐺 or denoted by 𝑙𝑚𝑑𝑠(𝐺). In this paper we explored the central local metric dimension on generalized fan graph, generalized broken fan graph, and a graph resulting from corona operation Cm ⊙ K¯n. The result show that the central local metric dimension of generalized fan graph is equal with its order because the diameter and radius are equal. Different with it, the central local metric dimension of generalized broken fan is equal with its local metric dimension plus cardinality of central set, and the central local metric dimension of Cm ⊙ K¯n are equal with the central local metric dimension of 𝐶𝑚.

A vertex \(k\)-labeling \(\phi:V(G)\rightarrow\{1,2,\dots,k\}\) on a simple graph \(G\) is said to be a distance irregular vertex \(k\)-labeling of \(G\) if the weights of all vertices of \(G\) are pairwise distinct, where the weight of a vertex is the sum of labels of all vertices adjacent to that vertex in \(G\). The least integer \(k\) for which \(G\) has a distance irregular vertex \(k\)-labeling is called the distance irregularity strength of \(G\) and denoted by \(\mathrm{dis}(G)\). In this paper, we introduce a new lower bound of distance irregularity strength of graphs and provide its sharpness for some graphs with pendant vertices. Moreover, some properties on distance irregularity strength for trees are also discussed in this paper.

A function ϕ: V(G)→{1, 2, …, k} of a simple graph G is said to be a non-inclusive distance vertex irregular k-labeling of G if the sums of labels of vertices in the open neighborhood of every vertex are distinct and is said to be an inclusive distance vertex irregular k-labeling of G if the sums of labels of vertices in the closed neighborhood of each vertex are different. The minimum k for which G has a non-inclusive (resp. an inclusive) distance vertex irregular k-labeling is called a non-inclusive (resp. an inclusive) distance irregularity strength and is denoted by dis(G) (resp. by dis(G)). In this paper, the non-inclusive and inclusive distance irregularity strength for the join product graphs are investigated.

Background:. Plastic and reconstructive surgery (PRS) academic positions are more coveted each year. We aim to determine the requirement of fellowship training before PRS academic appointments. Methods:. PRS faculty at U.S. academic institutions associated with the American Society of Plastic Surgeons were identified. Outcomes studied included integrated versus independent training, fellowships, gender, academic title, years on faculty, and publications before current hire. Results:. Of the 1052 PRS faculty identified, 646 were included across 41 states and the District of Columbia. Seventy-four percent were identified as men (n = 477), and 26.2% (n = 169) identified as women. Academic faculty were significantly more likely to have completed fellowship before hire than not (p

SuperH-antimagic total labeling (SHATL) can be utilized to generate encryption keys. The keys are then used to establish the improved block and stream ciphers. In these ciphers, different blocks were encrypted by the different keys, but all block keys were connected one another. These conditions make the developed cryptosystems more secure and require less keys storage capacity compared to the ordinary block and stream cipher. The randomness of the generated keys, however, still need to be tested. The test is necessary to ensure that there is no specific pattern that can be utilized by any intruder to guess the keys. This paper presents the randomness tests applied to all key sequences generated by both the improved block scheme and the stream based scheme.

Crowdfunding Platform (CFP) is another financing model that is different from the traditional financial institutions. CFP serves not only as a fundraising tool but also as a source of knowledge. CFP has several forms, namely equity-based, lending-based, reward-based, and donation-based. These forms of CFP divided into two classes, investment-based and non-investment-based. CFP is one form of platform Open Innovation (OI), where multiple stakeholders are involved in a funded project. OI refers to the collaborative innovation process. The involvement of multiple parties in OI leads to various risks that can result in collaboration failure, thus preventing the achievement of established CFP goals. Therefore, there is a need for effective governance in OI to manage potential conflicts, coordinate efficient knowledge sharing among the involved parties, and achieve the main objectives of the collaboration.

A function f with domain and range are respectively the edge set of graph G and natural number up to ke, and a function f with domain and range are respectively the vertex set of graph G and the even natural number up to 2kv are called a total k-labeling where k=max{ke,2kv}. The total k-labeling of graph G by the condition that every two different edges have different weight is called an edge irregular reflexive k-labeling, where for any edge x1x2, the weight is wt(x1x2)=fv(x1)+fe(x1x2)+fv(x2). The reflexive edge strength of the graph G, denoted by res(G) is the minimum k for graph G which has an edge irregular reflexive k-labelling. In this study, we obtained the res(G) of graphs which their vertex degrees show an almost regularity properties.

Background Although the novel 2019 coronavirus (COVID-19) pandemic delayed elective microsurgical reconstruction cases, traumatic injuries requiring microsurgical reconstruction for limb salvage still presented to our hospital. Recent data suggests a link between COVID-19 and increased venous and arterial thromboembolic events, theoretically increasing the risk of microvascular anastomotic thrombosis and flap failure. Methods We provide a single case report of microsurgical limb salvage in a COVID-19-positive patient with a Gustilo IIIb open tibial-fibular fracture. Results A free latissimus dorsi muscle flap was used to successfully cover a Gustilo IIIb open tibial-fibular fracture with exposed orthopedic hardware and large soft tissue defect, with 5-month follow-up demonstrating a well-healed flap allowing for preserved ambulation. Conclusion Although there is a theoretical increased incidence of venous and arterial thrombotic microvascular failure in COVID-19-positive patients, microsurgical reconstruction is occasionally the lone option for salvage. Microsurgical reconstruction in this population must be carefully considered, meticulously executed, and closely monitored.

The G be a connected graph with vertex set V(G) and edge set E(G). A subset S⊆V(G) is called a dominating set of G if for every vertex x in V(G)∖S, there exists at least one vertex u in S such that x is adjacent to u. An ordered set W⊆V(G) is called a resolving set of G, if every pair of vertices u and v in V(G) have distinct representation with respect to W. An ordered set S⊆V(G) is called a dominant resolving set of G, if S is a resolving set and also a dominating set of G. The minimum cardinality of dominant resolving set is called a dominant metric dimension of G, denoted by Ddim(G). In this paper, we investigate the dominant metric dimension of some particular class of graphs, the characterisation of graph with certain dominant metric dimension, and the dominant metric dimension of joint and comb products of graphs.

The local antimagic labeling on a graph G with |V| vertices and |E| edges is defined to be an assignment f : E --> {1, 2,..., |E|} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u)̸ ̸= w(v) where w(u) = Σe∈E(u) f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.

For a simple graph G, a vertex labeling f : V(G) → {1, 2, ..., k} is called a k-labeling. The weight of a vertex v, denoted by wtf(v) is the sum of all vertex labels of vertices in the closed neighborhood of the vertex v. A vertex k-labeling is defined to be an inclusive distance vertex irregular distance k-labeling of G if for every two different vertices u and v there is wtf(u) ≠ wtf(v). The minimum k for which the graph G has a vertex irregular distance k-labeling is called the inclusive distance vertex irregularity strength of G. In this paper we establish a lower bound of the inclusive distance vertex irregularity strength for any graph and determine the exact value of this parameter for several families of graphs.

The local resolving dominating set studied in this paper is a notion that combines two concepts in graph theory, the local metric dimension and dominations in graphs. Let G and H be connected graphs of orders n and m, respectively; and x a vertex in H hereafter referred to as a linkage vertex. The comb product of G and H denoted by G.H, is a graph obtained by taking one copy of G and n copies of H and attaching the i-th copy of H at the vertex x to the i-th vertex of G. In this paper, we determine the local resolving dominating set of the comb products G. Sn with two different linkage vertices, G. Kn, G. Km;n and G. Cm graphs. [ABSTRACT FROM AUTHOR]

Let G be a connected graph with vertex set V(G) and W={w1, w2, ..., wm} ⊆ V(G). A representation of a vertex v ∈ V(G) with respect to W is an ordered m-tuple r(v|W)=(d(v,w1),d(v,w2),...,d(v,wm)) where d(v,w) is the distance between vertices v and w. The set W is called a resolving set for G if every vertex of G has a distinct representation with respect to W. A resolving set containing a minimum number of vertices is called a basis for G. The metric dimension of G, denoted by dim (G), is the number of vertices in a basis of G. In general, the comb product and the corona product are non-commutative operations in a graph. However, these operations can be commutative with respect to the metric dimension for some graphs with certain conditions. In this paper, we determine the metric dimension of the generalized comb and corona products of graphs and the necessary and sufficient conditions of the graphs in order for the comb and corona products to be commutative operations with respect to the metric dimension.