Research on the independence polynomial of graphs has been very active.However,the computational complexity of determining independence polynomials for general graphs remains NP-hard.Letα(G)be the independence number...Research on the independence polynomial of graphs has been very active.However,the computational complexity of determining independence polynomials for general graphs remains NP-hard.Letα(G)be the independence number of G and i(G;k)be the number of independent sets of order k in G,then the independence polynomial is defined as I(G;x)=∑_(k=0)^(α(G))i(G;k)x^(k),i(G;0)=1.In this paper,by utilizing the transfer matrix,we obtain an analytical expression for I(CGn;x)of mono-cylindrical grid graphs CGn and present a crucial proof of it.Moreover,we also explore the Merrifield-Simmons index and other properties of CGn.展开更多
A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality...A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by <img src="Edit_82259359-0135-4a65-9378-b767f0405b48.png" alt="" />. A set <em>C ⊆ V (G)</em> is called a chromatic set if <em>C</em> contains all vertices of different colors in<em> G</em>, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by <img src="Edit_d849148d-5778-459b-abbb-ff25b5cd659b.png" alt="" />. A geo-chromatic set<em> S</em><sub><em>c</em></sub><em> ⊆ V (G</em><em>)</em> is both a geodetic set and a chromatic set. The geo-chromatic number <img src="Edit_505e203c-888c-471c-852d-4b9c2dd1a31c.png" alt="" /><em> </em>of<em> G</em> is the minimum cardinality among all geo-chromatic sets of<em> G</em>. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.展开更多
A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast dominat...A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength t that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least r from its set of broadcasting neighbors. In this paper, we extend the study of (t, r) broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed (t, r) broadcast domination numbers of all orientations of a graph G. In particular, we prove that in the cases r = 1 and (t, r) = (2, 2), for every integer value in this interval, there exists an orientation of G which has directed (t, r) broadcast domination number equal to that value. We also investigate directed (t, r) broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.展开更多
Let γ f(G) and γ~t f(G) be the fractional domination number and fractional total domination number of a graph G respectively. Hare and Stewart gave some exact fractional domination number of P n...Let γ f(G) and γ~t f(G) be the fractional domination number and fractional total domination number of a graph G respectively. Hare and Stewart gave some exact fractional domination number of P n×P m (grid graph) with small n and m . But for large n and m , it is difficult to decide the exact fractional domination number. Motivated by this, nearly sharp upper and lower bounds are given to the fractional domination number of grid graphs. Furthermore, upper and lower bounds on the fractional total domination number of strong direct product of graphs are given.展开更多
Let G(V, E) be a finite connected simple graph with vertex set V(G). A function is a signed dominating function f : <em style="white-space:normal;">V<span style="white-space:normal;"&...Let G(V, E) be a finite connected simple graph with vertex set V(G). A function is a signed dominating function f : <em style="white-space:normal;">V<span style="white-space:normal;">(<em style="white-space:normal;">G<span style="white-space:normal;">)<span style="white-space:nowrap;">→{<span style="white-space:nowrap;"><span style="white-space:nowrap;">−1,1} if for every vertex v <span style="white-space:nowrap;">∈ V(G), the sum of closed neighborhood weights of v is greater or equal to 1. The signed domination number γ<sub>s</sub>(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths P<sub>m</sub> and P<sub>n</sub> for m = 6, 7 and arbitrary n.展开更多
By modeling the spatiotemporal data of the power grid, it is possible to better understand its operational status, identify potential issues and risks, and take timely measures to adjust and optimize the system. Compa...By modeling the spatiotemporal data of the power grid, it is possible to better understand its operational status, identify potential issues and risks, and take timely measures to adjust and optimize the system. Compared to the bus-branch model, the node-breaker model provides higher granularity in describing grid components and can dynamically reflect changes in equipment status, thus improving the efficiency of grid dispatching and operation. This paper proposes a spatiotemporal data modeling method based on a graph database. It elaborates on constructing graph nodes, graph ontology models, and graph entity models from grid dispatch data, describing the construction of the spatiotemporal node-breaker graph model and the transformation to the bus-branch model. Subsequently, by integrating spatiotemporal data attributes into the pre-built static grid graph model, a spatiotemporal evolving graph of the power grid is constructed. Furthermore, the concept of the “Power Grid One Graph” and its requirements in modern power systems are elucidated. Leveraging the constructed spatiotemporal node-breaker graph model and graph computing technology, the paper explores the feasibility of grid situational awareness. Finally, typical applications in an operational provincial grid are showcased, and potential scenarios of the proposed spatiotemporal graph model are discussed.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.U20A20228)Huzhou Science and Technology Plan Project(Grant No.2022YZ53).
文摘Research on the independence polynomial of graphs has been very active.However,the computational complexity of determining independence polynomials for general graphs remains NP-hard.Letα(G)be the independence number of G and i(G;k)be the number of independent sets of order k in G,then the independence polynomial is defined as I(G;x)=∑_(k=0)^(α(G))i(G;k)x^(k),i(G;0)=1.In this paper,by utilizing the transfer matrix,we obtain an analytical expression for I(CGn;x)of mono-cylindrical grid graphs CGn and present a crucial proof of it.Moreover,we also explore the Merrifield-Simmons index and other properties of CGn.
文摘A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by <img src="Edit_82259359-0135-4a65-9378-b767f0405b48.png" alt="" />. A set <em>C ⊆ V (G)</em> is called a chromatic set if <em>C</em> contains all vertices of different colors in<em> G</em>, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by <img src="Edit_d849148d-5778-459b-abbb-ff25b5cd659b.png" alt="" />. A geo-chromatic set<em> S</em><sub><em>c</em></sub><em> ⊆ V (G</em><em>)</em> is both a geodetic set and a chromatic set. The geo-chromatic number <img src="Edit_505e203c-888c-471c-852d-4b9c2dd1a31c.png" alt="" /><em> </em>of<em> G</em> is the minimum cardinality among all geo-chromatic sets of<em> G</em>. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.
文摘A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength t that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least r from its set of broadcasting neighbors. In this paper, we extend the study of (t, r) broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed (t, r) broadcast domination numbers of all orientations of a graph G. In particular, we prove that in the cases r = 1 and (t, r) = (2, 2), for every integer value in this interval, there exists an orientation of G which has directed (t, r) broadcast domination number equal to that value. We also investigate directed (t, r) broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.
文摘Let γ f(G) and γ~t f(G) be the fractional domination number and fractional total domination number of a graph G respectively. Hare and Stewart gave some exact fractional domination number of P n×P m (grid graph) with small n and m . But for large n and m , it is difficult to decide the exact fractional domination number. Motivated by this, nearly sharp upper and lower bounds are given to the fractional domination number of grid graphs. Furthermore, upper and lower bounds on the fractional total domination number of strong direct product of graphs are given.
文摘Let G(V, E) be a finite connected simple graph with vertex set V(G). A function is a signed dominating function f : <em style="white-space:normal;">V<span style="white-space:normal;">(<em style="white-space:normal;">G<span style="white-space:normal;">)<span style="white-space:nowrap;">→{<span style="white-space:nowrap;"><span style="white-space:nowrap;">−1,1} if for every vertex v <span style="white-space:nowrap;">∈ V(G), the sum of closed neighborhood weights of v is greater or equal to 1. The signed domination number γ<sub>s</sub>(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths P<sub>m</sub> and P<sub>n</sub> for m = 6, 7 and arbitrary n.
基金supported by the Project of China Southern Power Grid Digital Grid Research Institute Co.,Ltd.(210002KK52222026)。
文摘By modeling the spatiotemporal data of the power grid, it is possible to better understand its operational status, identify potential issues and risks, and take timely measures to adjust and optimize the system. Compared to the bus-branch model, the node-breaker model provides higher granularity in describing grid components and can dynamically reflect changes in equipment status, thus improving the efficiency of grid dispatching and operation. This paper proposes a spatiotemporal data modeling method based on a graph database. It elaborates on constructing graph nodes, graph ontology models, and graph entity models from grid dispatch data, describing the construction of the spatiotemporal node-breaker graph model and the transformation to the bus-branch model. Subsequently, by integrating spatiotemporal data attributes into the pre-built static grid graph model, a spatiotemporal evolving graph of the power grid is constructed. Furthermore, the concept of the “Power Grid One Graph” and its requirements in modern power systems are elucidated. Leveraging the constructed spatiotemporal node-breaker graph model and graph computing technology, the paper explores the feasibility of grid situational awareness. Finally, typical applications in an operational provincial grid are showcased, and potential scenarios of the proposed spatiotemporal graph model are discussed.
