Pub Date : 2023-09-29DOI: 10.1142/s0219265923500184
G. Mohan Ram, E. Ilavarasan
In wireless sensor networks (WSN), the multifunctional low-power sensor nodes are linked to base stations and are highly responsible for sensing various information. However, maintaining the network’s lifespan is a major issue because of limited battery capacity and node failures. Thus, the proposed study introduced a new Scalable Trust-based Energy Efficient Routing Algorithm (STEERA) with Adaptive Grasshopper Algorithm (AGA) to enhance the network lifespan. At first, the trust values (Tv) are generated for each node to eliminate the malicious node with the help of penalty factor and volatilization. To make efficient routing process, the nodes are formed into cluster groups. A suitable cluster head is selected for each cluster to mitigate the energy consumption problem through the AGA approach by optimizing the parameters like Tv, distance and residual energy. Finally, the data packets are effectively transmitted through a multi-hop routing process by selecting an appropriate path and satisfying certain parameters through Fire Hawks Optimization (FHO). To simulate the proposed techniques, NS2 software is employed, and the performance is measured over various metrics like energy consumption, residual energy, the lifespan of a network, packet loss ratio and average end-to-end delay. The performance analysis shows that the proposed protocol design has obtained higher results than conventional routing protocols.
{"title":"Secure and Energy-Based STEERA Routing Protocol for Wireless Sensor Networks","authors":"G. Mohan Ram, E. Ilavarasan","doi":"10.1142/s0219265923500184","DOIUrl":"https://doi.org/10.1142/s0219265923500184","url":null,"abstract":"In wireless sensor networks (WSN), the multifunctional low-power sensor nodes are linked to base stations and are highly responsible for sensing various information. However, maintaining the network’s lifespan is a major issue because of limited battery capacity and node failures. Thus, the proposed study introduced a new Scalable Trust-based Energy Efficient Routing Algorithm (STEERA) with Adaptive Grasshopper Algorithm (AGA) to enhance the network lifespan. At first, the trust values (Tv) are generated for each node to eliminate the malicious node with the help of penalty factor and volatilization. To make efficient routing process, the nodes are formed into cluster groups. A suitable cluster head is selected for each cluster to mitigate the energy consumption problem through the AGA approach by optimizing the parameters like Tv, distance and residual energy. Finally, the data packets are effectively transmitted through a multi-hop routing process by selecting an appropriate path and satisfying certain parameters through Fire Hawks Optimization (FHO). To simulate the proposed techniques, NS2 software is employed, and the performance is measured over various metrics like energy consumption, residual energy, the lifespan of a network, packet loss ratio and average end-to-end delay. The performance analysis shows that the proposed protocol design has obtained higher results than conventional routing protocols.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135243461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-26DOI: 10.1142/s0219265923500196
Wayne Goddard, Julia VanLandingham
The integrity of a graph [Formula: see text] is defined as the minimum value of [Formula: see text] taken over all [Formula: see text], where [Formula: see text] denotes the maximum cardinality of a component of graph [Formula: see text]. In this paper, we investigate bounds on the maximum and minimum values of the weighted version of this parameter. We also consider the same question for the related parameter vertex-neighbor-integrity.
{"title":"On the Extremal Values of the Weighted Integrity of a Graph","authors":"Wayne Goddard, Julia VanLandingham","doi":"10.1142/s0219265923500196","DOIUrl":"https://doi.org/10.1142/s0219265923500196","url":null,"abstract":"The integrity of a graph [Formula: see text] is defined as the minimum value of [Formula: see text] taken over all [Formula: see text], where [Formula: see text] denotes the maximum cardinality of a component of graph [Formula: see text]. In this paper, we investigate bounds on the maximum and minimum values of the weighted version of this parameter. We also consider the same question for the related parameter vertex-neighbor-integrity.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135720970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-26DOI: 10.1142/s0219265923500172
Dongjuan Ma, Feng Jing, Zehui Liu, Min Guo, Weizhe Jing, Jie Liang
Power information data transmission is prone to packet loss rate, and communication protocol is required to improve the transmission capacity of power information data. Therefore, a low delay transmission scheme of power communication information based on 5G networking is proposed and constructed, the power communication information data through 5G networking technology is processed and is sent to the corresponding data application module for data classification and mining, and then the information data is converted in different formats to the same data transmission format. TCP communication protocol as the communication protocol is selected for data online transmission to describe the low delay transmission of power communication information. The chaotic sequence is set based on 5G networking, and the connection weight matrix is calculated. Then the power communication information sequence is encrypted in real time according to the row column substitution rule. On this basis, the data encryption module, dynamic key generation module and shared key update module are combined to realize the low delay transmission of power communication information. The experimental results show that with the increase of the network area and the number of nodes, the data transmission volume can be gradually increased. The change range of this method is relatively small, which can effectively improve the feasibility of the low delay transmission scheme of power communication information.
{"title":"Low Delay Transmission Scheme of Power Communication Information Based on 5G Network","authors":"Dongjuan Ma, Feng Jing, Zehui Liu, Min Guo, Weizhe Jing, Jie Liang","doi":"10.1142/s0219265923500172","DOIUrl":"https://doi.org/10.1142/s0219265923500172","url":null,"abstract":"Power information data transmission is prone to packet loss rate, and communication protocol is required to improve the transmission capacity of power information data. Therefore, a low delay transmission scheme of power communication information based on 5G networking is proposed and constructed, the power communication information data through 5G networking technology is processed and is sent to the corresponding data application module for data classification and mining, and then the information data is converted in different formats to the same data transmission format. TCP communication protocol as the communication protocol is selected for data online transmission to describe the low delay transmission of power communication information. The chaotic sequence is set based on 5G networking, and the connection weight matrix is calculated. Then the power communication information sequence is encrypted in real time according to the row column substitution rule. On this basis, the data encryption module, dynamic key generation module and shared key update module are combined to realize the low delay transmission of power communication information. The experimental results show that with the increase of the network area and the number of nodes, the data transmission volume can be gradually increased. The change range of this method is relatively small, which can effectively improve the feasibility of the low delay transmission scheme of power communication information.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135720666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-10DOI: 10.1142/s0219265923500160
Caixi Xue, Shuming Zhou, Hong Zhang
Connectivity along with its extensions are important metrices to estimate the fault-tolerance of interconnection networks. The classic connectivity [Formula: see text] of a graph [Formula: see text] is the minimum cardinality of a vertex set [Formula: see text] such that [Formula: see text] is connected or a single vertex. For any subset [Formula: see text] with [Formula: see text], a tree [Formula: see text] in [Formula: see text] is called an [Formula: see text]-tree if [Formula: see text]. Furthermore, any two [Formula: see text]-tree [Formula: see text] and [Formula: see text] are internally disjoint if [Formula: see text] and [Formula: see text]. We denote by [Formula: see text] the maximum number of pairwise internally disjoint [Formula: see text]-trees in [Formula: see text]. For an integer [Formula: see text], the generalized [Formula: see text]-connectivity of a graph [Formula: see text] is defined as [Formula: see text] and [Formula: see text]. For the [Formula: see text]-dimensional folded divide-and-swap cubes, [Formula: see text], we show the upper bound and the lower bound of [Formula: see text], that is [Formula: see text], where [Formula: see text] and [Formula: see text] in this paper.
{"title":"The Bounds of Generalized 4-Connectivity of Folded Divide-and-Swap Cubes","authors":"Caixi Xue, Shuming Zhou, Hong Zhang","doi":"10.1142/s0219265923500160","DOIUrl":"https://doi.org/10.1142/s0219265923500160","url":null,"abstract":"Connectivity along with its extensions are important metrices to estimate the fault-tolerance of interconnection networks. The classic connectivity [Formula: see text] of a graph [Formula: see text] is the minimum cardinality of a vertex set [Formula: see text] such that [Formula: see text] is connected or a single vertex. For any subset [Formula: see text] with [Formula: see text], a tree [Formula: see text] in [Formula: see text] is called an [Formula: see text]-tree if [Formula: see text]. Furthermore, any two [Formula: see text]-tree [Formula: see text] and [Formula: see text] are internally disjoint if [Formula: see text] and [Formula: see text]. We denote by [Formula: see text] the maximum number of pairwise internally disjoint [Formula: see text]-trees in [Formula: see text]. For an integer [Formula: see text], the generalized [Formula: see text]-connectivity of a graph [Formula: see text] is defined as [Formula: see text] and [Formula: see text]. For the [Formula: see text]-dimensional folded divide-and-swap cubes, [Formula: see text], we show the upper bound and the lower bound of [Formula: see text], that is [Formula: see text], where [Formula: see text] and [Formula: see text] in this paper.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83303352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-08DOI: 10.1142/s0219265923500147
Merlin Thomas Ellumkalayil, S. Naduvath
A graph coloring is proper when the colors assigned to a pair of adjacent vertices in it are different and it is improper when at least one of the adjacent pair of vertices receives the same color. When the minimum number of colors required in a proper coloring of a graph is not available, coloring the graph with the available colors, say [Formula: see text] colors, will lead at least an edge to have its end vertices colored with a same color. Such an edge is called a bad edge. In a proper coloring of a graph [Formula: see text], every color class is an independent set. However, in an improper coloring there can be few color classes that are non-independent. In this paper, we use the concept of [Formula: see text]-coloring, which permits only one color class to be non-independent and determine the minimum number of bad edges, which is denoted by [Formula: see text], obtained from the same for some cycle-related graphs.
{"title":"On δ(k)-Coloring of Some Cycle-Related Graphs","authors":"Merlin Thomas Ellumkalayil, S. Naduvath","doi":"10.1142/s0219265923500147","DOIUrl":"https://doi.org/10.1142/s0219265923500147","url":null,"abstract":"A graph coloring is proper when the colors assigned to a pair of adjacent vertices in it are different and it is improper when at least one of the adjacent pair of vertices receives the same color. When the minimum number of colors required in a proper coloring of a graph is not available, coloring the graph with the available colors, say [Formula: see text] colors, will lead at least an edge to have its end vertices colored with a same color. Such an edge is called a bad edge. In a proper coloring of a graph [Formula: see text], every color class is an independent set. However, in an improper coloring there can be few color classes that are non-independent. In this paper, we use the concept of [Formula: see text]-coloring, which permits only one color class to be non-independent and determine the minimum number of bad edges, which is denoted by [Formula: see text], obtained from the same for some cycle-related graphs.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81097134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-08DOI: 10.1142/s0219265923500159
Mahmoud Mezghani, Omnia Mezghani
In this paper, we propose an approach solving the holes problem in Wireless Sensor Networks (WSNs) based on Khalimsky k-Clustering and data routing protocol (MDKC). The aim of this solution is to establish optimized data routing paths between isolated nodes/clusters and the Sink in noisy environment with the presence of obstacles. This approach is an improvement of a previous work deploying stationary WSN not dealing with the problem of “holes”. At first, the MDKC algorithm divides the WSN into k-hop [Formula: see text] compact dynamic clusters. For each cluster, a node is elected cluster-head in its k-neighborhood according to some criteria such as the remaining energy, the k-degree and the communication probability average. Then, some nodes are selected as Khalimsky anchors to optimize the intra-cluster data routing process. The Khalimsky anchors at the border layers ensure the inter-cluster data routing between adjacent clusters. In the next phase of MDKC, Mobile Collectors (MCs) are used for data collecting and relaying from isolated nodes/clusters to the connected Khalimsky’s nodes. The simulation results prove that MDKC minimizes the energy consumption and improves the connectivity rate between sensors and the delivery rate compared to some existing approaches.
{"title":"Holes Problem Solving in Khalimsky Topology Protocol for Wireless Sensor Networks (WSNs)","authors":"Mahmoud Mezghani, Omnia Mezghani","doi":"10.1142/s0219265923500159","DOIUrl":"https://doi.org/10.1142/s0219265923500159","url":null,"abstract":"In this paper, we propose an approach solving the holes problem in Wireless Sensor Networks (WSNs) based on Khalimsky k-Clustering and data routing protocol (MDKC). The aim of this solution is to establish optimized data routing paths between isolated nodes/clusters and the Sink in noisy environment with the presence of obstacles. This approach is an improvement of a previous work deploying stationary WSN not dealing with the problem of “holes”. At first, the MDKC algorithm divides the WSN into k-hop [Formula: see text] compact dynamic clusters. For each cluster, a node is elected cluster-head in its k-neighborhood according to some criteria such as the remaining energy, the k-degree and the communication probability average. Then, some nodes are selected as Khalimsky anchors to optimize the intra-cluster data routing process. The Khalimsky anchors at the border layers ensure the inter-cluster data routing between adjacent clusters. In the next phase of MDKC, Mobile Collectors (MCs) are used for data collecting and relaying from isolated nodes/clusters to the connected Khalimsky’s nodes. The simulation results prove that MDKC minimizes the energy consumption and improves the connectivity rate between sensors and the delivery rate compared to some existing approaches.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81677002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-27DOI: 10.1142/s0219265923500135
Zhiqiang Ma, Fengyun Ren, Meiqin Wei, Gemaji Bao
For a family of connected graphs [Formula: see text], a spanning subgraph [Formula: see text] of [Formula: see text] is called an [Formula: see text]-factor if each component of [Formula: see text] is isomorphic to a member of [Formula: see text]. In this paper, sufficient conditions with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the [Formula: see text]-factor, [Formula: see text]-factor or [Formula: see text]-factor for [Formula: see text] are obtained.
{"title":"Sufficient Conditions of (Isolated) Toughness and Binding Number for the Existence of Component Factors","authors":"Zhiqiang Ma, Fengyun Ren, Meiqin Wei, Gemaji Bao","doi":"10.1142/s0219265923500135","DOIUrl":"https://doi.org/10.1142/s0219265923500135","url":null,"abstract":"For a family of connected graphs [Formula: see text], a spanning subgraph [Formula: see text] of [Formula: see text] is called an [Formula: see text]-factor if each component of [Formula: see text] is isomorphic to a member of [Formula: see text]. In this paper, sufficient conditions with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the [Formula: see text]-factor, [Formula: see text]-factor or [Formula: see text]-factor for [Formula: see text] are obtained.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90853715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-19DOI: 10.1142/s0219265923500093
Xue Feng, Shenglin Geng, Banghe Han
An efficient solution to the misleading solutions of decision-making problems is the study of consistency when the decision makers express their opinions by means of fuzzy preference relations. To elucidate the consistency of HFPRs, the S-ordinal consistency of HFPRs was proposed, and S-OCI was also proposed to evaluate the degree of consistency of a hesitant fuzzy preference relation by calculating the unreasonable 3-cycles in the directed graph. Two novel methods were also proposed for calculation of the S-OCI. Moreover, the inconsistent judgment in hesitation fuzzy preference relation was developed. In order to repair the inconsistency of a hesitant fuzzy preference relation, an algorithm for finding and removing 3-cycles in digraph was developed. Finally, some illustrative examples were given to prove the effectiveness of the method.
{"title":"The Ordinal Consistency of a Hesitant Fuzzy Preference Relation","authors":"Xue Feng, Shenglin Geng, Banghe Han","doi":"10.1142/s0219265923500093","DOIUrl":"https://doi.org/10.1142/s0219265923500093","url":null,"abstract":"An efficient solution to the misleading solutions of decision-making problems is the study of consistency when the decision makers express their opinions by means of fuzzy preference relations. To elucidate the consistency of HFPRs, the S-ordinal consistency of HFPRs was proposed, and S-OCI was also proposed to evaluate the degree of consistency of a hesitant fuzzy preference relation by calculating the unreasonable 3-cycles in the directed graph. Two novel methods were also proposed for calculation of the S-OCI. Moreover, the inconsistent judgment in hesitation fuzzy preference relation was developed. In order to repair the inconsistency of a hesitant fuzzy preference relation, an algorithm for finding and removing 3-cycles in digraph was developed. Finally, some illustrative examples were given to prove the effectiveness of the method.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75870466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-19DOI: 10.1142/s0219265923500123
Guoqiang Xie, J. Meng
A strong digraph [Formula: see text] is strongly Menger (arc) connected if, for [Formula: see text], [Formula: see text] can reach [Formula: see text] by min[Formula: see text] internally (arc) disjoint-directed paths. A digraph [Formula: see text] is [Formula: see text](-arc)-fault-tolerant strongly Menger([Formula: see text]-(A)FTSM, for short) (arc) connected if [Formula: see text] is strongly Menger (arc) connected for every [Formula: see text](respectively, [Formula: see text]) with [Formula: see text]. A digraph [Formula: see text] is [Formula: see text]-conditional (arc)-fault-tolerant strongly Menger ([Formula: see text]-C(A)FTSM, for short) (arc) connected if [Formula: see text] is strongly Menger (arc) connected for every [Formula: see text](respectively, [Formula: see text]) with [Formula: see text] and [Formula: see text]. The directed [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] [Formula: see text] and [Formula: see text] is a digraph with vertex set [Formula: see text]. For two vertices [Formula: see text] and [Formula: see text], [Formula: see text] dominates [Formula: see text] if there exists an integer [Formula: see text], [Formula: see text], satisfying [Formula: see text]mod [Formula: see text] and [Formula: see text], when [Formula: see text]. In this paper, we show that [Formula: see text] [Formula: see text] is [Formula: see text]-AFTSM arc connected when [Formula: see text], [Formula: see text]-FTSM connected when [Formula: see text], [Formula: see text]-CAFTSM arc connected when [Formula: see text], and [Formula: see text]-CFTSM connected when [Formula: see text].
{"title":"The Strong Menger Connectivity of the Directed k-Ary n-Cube","authors":"Guoqiang Xie, J. Meng","doi":"10.1142/s0219265923500123","DOIUrl":"https://doi.org/10.1142/s0219265923500123","url":null,"abstract":"A strong digraph [Formula: see text] is strongly Menger (arc) connected if, for [Formula: see text], [Formula: see text] can reach [Formula: see text] by min[Formula: see text] internally (arc) disjoint-directed paths. A digraph [Formula: see text] is [Formula: see text](-arc)-fault-tolerant strongly Menger([Formula: see text]-(A)FTSM, for short) (arc) connected if [Formula: see text] is strongly Menger (arc) connected for every [Formula: see text](respectively, [Formula: see text]) with [Formula: see text]. A digraph [Formula: see text] is [Formula: see text]-conditional (arc)-fault-tolerant strongly Menger ([Formula: see text]-C(A)FTSM, for short) (arc) connected if [Formula: see text] is strongly Menger (arc) connected for every [Formula: see text](respectively, [Formula: see text]) with [Formula: see text] and [Formula: see text]. The directed [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] [Formula: see text] and [Formula: see text] is a digraph with vertex set [Formula: see text]. For two vertices [Formula: see text] and [Formula: see text], [Formula: see text] dominates [Formula: see text] if there exists an integer [Formula: see text], [Formula: see text], satisfying [Formula: see text]mod [Formula: see text] and [Formula: see text], when [Formula: see text]. In this paper, we show that [Formula: see text] [Formula: see text] is [Formula: see text]-AFTSM arc connected when [Formula: see text], [Formula: see text]-FTSM connected when [Formula: see text], [Formula: see text]-CAFTSM arc connected when [Formula: see text], and [Formula: see text]-CFTSM connected when [Formula: see text].","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90673121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-17DOI: 10.1142/s0219265923500111
Ying-zi Ma, Hui Zhang
Given a set [Formula: see text] with [Formula: see text], a tree [Formula: see text] is considered as a total proper [Formula: see text]-tree or a total proper tree connecting [Formula: see text] if any two adjacent or incident elements of edges and [Formula: see text] of [Formula: see text] differ in color. Let [Formula: see text] be a connected graph of order [Formula: see text], and [Formula: see text] be an integer with [Formula: see text]. A total-colored graph is total proper[Formula: see text]-tree connected if for every set [Formula: see text] of [Formula: see text] vertices, there exists a total proper [Formula: see text]-tree in [Formula: see text]. The [Formula: see text]-total-proper index of [Formula: see text], denoted by [Formula: see text], is the minimum number of colors required to make [Formula: see text] total proper [Formula: see text]-tree connected. In this paper, we first investigate the [Formula: see text]-total-proper index of some special graphs. Moreover, we characterize the graphs with [Formula: see text]-total-proper index [Formula: see text] and [Formula: see text], respectively.
{"title":"The k-Total-Proper Index of Graphs","authors":"Ying-zi Ma, Hui Zhang","doi":"10.1142/s0219265923500111","DOIUrl":"https://doi.org/10.1142/s0219265923500111","url":null,"abstract":"Given a set [Formula: see text] with [Formula: see text], a tree [Formula: see text] is considered as a total proper [Formula: see text]-tree or a total proper tree connecting [Formula: see text] if any two adjacent or incident elements of edges and [Formula: see text] of [Formula: see text] differ in color. Let [Formula: see text] be a connected graph of order [Formula: see text], and [Formula: see text] be an integer with [Formula: see text]. A total-colored graph is total proper[Formula: see text]-tree connected if for every set [Formula: see text] of [Formula: see text] vertices, there exists a total proper [Formula: see text]-tree in [Formula: see text]. The [Formula: see text]-total-proper index of [Formula: see text], denoted by [Formula: see text], is the minimum number of colors required to make [Formula: see text] total proper [Formula: see text]-tree connected. In this paper, we first investigate the [Formula: see text]-total-proper index of some special graphs. Moreover, we characterize the graphs with [Formula: see text]-total-proper index [Formula: see text] and [Formula: see text], respectively.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74470764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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