For the subset S■V(G), if every path with k vertices in a graph G contains at least one vertex from S, we call that S is a k-path vertex cover set of the graph G. Obviously, the subset is not unique. The cardinality ...For the subset S■V(G), if every path with k vertices in a graph G contains at least one vertex from S, we call that S is a k-path vertex cover set of the graph G. Obviously, the subset is not unique. The cardinality of the minimum k-path vertex cover set of a graph G is called the k-path vertex cover number, we denote it by ψk(G). In this paper, a lower or upper bound of ψk for some Cartesian product graphs is presented.展开更多
Outer-independent Roman domination on graphs originates from the defensive strategy of Ancient Rome,which is that if any city without an army is attacked,a neighboring city with two armies could mobilize an army to su...Outer-independent Roman domination on graphs originates from the defensive strategy of Ancient Rome,which is that if any city without an army is attacked,a neighboring city with two armies could mobilize an army to support it and any two cities that have no army cannot be adjacent.The outer-independent Roman domination on graphs is an attractive topic in graph theory,and the definition is described as follows.Given a graph G=(V,E),a function f:V(G)→{0,1,2}is an outer-independent Roman dominating function(OIRDF)if f satisfies that every vertex v∈V with f(v)=0 has at least one adjacent vertex u∈N(v)with f(u)=2,where N(v)is the open neighborhood of v,and the set V0={v|f(v)=0}is an independent set.The weight of an OIRDF f is w(f)=∑_(v∈V)f(v).The value of minf w(f)is the outerindependent Roman domination number of G,denoted asγoiR(G).This paper is devoted to the study of the outer-independent Roman domination number of the Cartesian product of paths P_(n)□P_(m).With the help of computer,we find some recursive OIRDFs and then we present an upper bound ofγoiR(P_(n)□P_(m)).Furthermore,we prove the lower bound ofγoiR(P_(n)□P_(m))(n≤3)is equal to the upper bound.Hence,we achieve the exact value ofγoiR(P_(n)□P_(m))for n≤3 and the upper bound ofγoiR(P_(n)□P_(m))for n≥4.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
The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. ...The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.展开更多
Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are...Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L（j, k）-labeling of G is called the circular L（j, k）-labeling number of G and is denoted by σj, k（G）. For any two positive integers j and k with j≤k,the circular L（j, k）-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined.展开更多
A set ?is a dominating set of G if every vertex of ?is adjacent to at least one vertex of S. The cardinality of the smallest dominating set of G is called the domination number of G. The square G2 of a graph G is obta...A set ?is a dominating set of G if every vertex of ?is adjacent to at least one vertex of S. The cardinality of the smallest dominating set of G is called the domination number of G. The square G2 of a graph G is obtained from G by adding new edges between every two vertices having distance 2 in G. In this paper we study the domination number of square of graphs, find a bound for domination number of square of Cartesian product of cycles, and find the exact value for some of them.展开更多
Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination numb...Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, ?(mod 3) and bounds for otherwise.展开更多
In this paper,we focus on inferring graph Laplacian matrix from the spatiotemporal signal which is defined as“time-vertex signal”.To realize this,we first represent the signals on a joint graph which is the Cartesia...In this paper,we focus on inferring graph Laplacian matrix from the spatiotemporal signal which is defined as“time-vertex signal”.To realize this,we first represent the signals on a joint graph which is the Cartesian product graph of the time-and vertex-graphs.By assuming the signals follow a Gaussian prior distribution on the joint graph,a meaningful representation that promotes the smoothness property of the joint graph signal is derived.Furthermore,by decoupling the joint graph,the graph learning framework is formulated as a joint optimization problem which includes signal denoising,timeand vertex-graphs learning together.Specifically,two algorithms are proposed to solve the optimization problem,where the discrete second-order difference operator with reversed sign(DSODO)in the time domain is used as the time-graph Laplacian operator to recover the signal and infer a vertex-graph in the first algorithm,and the time-graph,as well as the vertex-graph,is estimated by the other algorithm.Experiments on both synthetic and real-world datasets demonstrate that the proposed algorithms can effectively infer meaningful time-and vertex-graphs from noisy and incomplete data.展开更多
A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set ...A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].展开更多
In this paper we prove that the complete bipartite graph kmn where m and n are even, join of two cycle graphs cn and cm where n + m ≡ 0 (mod 4), split graph of cn for even “n”, Kn × P2 where n is even are admi...In this paper we prove that the complete bipartite graph kmn where m and n are even, join of two cycle graphs cn and cm where n + m ≡ 0 (mod 4), split graph of cn for even “n”, Kn × P2 where n is even are admits a Zero-M-Cordial labeling. Further we prove that Kn × P2Bn = K1,n × P2 of odd n admits a Zero-M-Cordial labeling.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
Let G =( V,E) be a connected graph and W = { w_1,w_2,…,w_k} be an ordered subset of V( G).For any vertex v ∈V,the locating code of v with respect to W is the k-vector CW( v) = { d( v,w_1),d( v,w_2),…,d( v,w_k) },W ...Let G =( V,E) be a connected graph and W = { w_1,w_2,…,w_k} be an ordered subset of V( G).For any vertex v ∈V,the locating code of v with respect to W is the k-vector CW( v) = { d( v,w_1),d( v,w_2),…,d( v,w_k) },W is said to be a locating set of G if distinct vertices have the distinct locating code,and the locating number of G is defined as: Loc( G) = min{ | W| : W is a locating set of G}.We study the locating set and locating number of a graph G,obtain some bounds for the locating numbers of graphs,and determine the exact value of Loc( G) for some special classes of graphs,such as cycles,wheels,complete t-partite graph and some Cartesian products of paths and cycles. In addition,we also prove that Loc( T) ≥Δ-1 holds for all trees T with maximum degree Δ,and shows a tree T with Loc( T) = Δ-1.展开更多
Abstract Let U(R,S) denote the class of all (0,1) m×n matrices having row sum vector R and column sum vector S. The interchange graph G(R,S) is the graph where the vertices are the matrices in U(R,S) and two...Abstract Let U(R,S) denote the class of all (0,1) m×n matrices having row sum vector R and column sum vector S. The interchange graph G(R,S) is the graph where the vertices are the matrices in U(R,S) and two vertices representing two such matrices are adjacent provided they differ by an interchange. It is proved that G(R,(1,1,...,1)) is a generalized Cartesian product of some Johnson Scheme graphs. Furthermore, its connectivity, diameter and transitivity (vertex ,edge ) are also determined.展开更多
In this paper, the crossing numbers of the Cartesian products of a specific 5-vertex graph with a star are given, and thus the result fills up the crossing number list of Cartesian products of all 5-vertex graphs with...In this paper, the crossing numbers of the Cartesian products of a specific 5-vertex graph with a star are given, and thus the result fills up the crossing number list of Cartesian products of all 5-vertex graphs with stars （presented by Marian Klesc）. In addition, we also give an up to date description of Cartesian products of 5-vertex graphs with stars, whose crossing numbers are known.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.61463026,61463027).
文摘For the subset S■V(G), if every path with k vertices in a graph G contains at least one vertex from S, we call that S is a k-path vertex cover set of the graph G. Obviously, the subset is not unique. The cardinality of the minimum k-path vertex cover set of a graph G is called the k-path vertex cover number, we denote it by ψk(G). In this paper, a lower or upper bound of ψk for some Cartesian product graphs is presented.
文摘Outer-independent Roman domination on graphs originates from the defensive strategy of Ancient Rome,which is that if any city without an army is attacked,a neighboring city with two armies could mobilize an army to support it and any two cities that have no army cannot be adjacent.The outer-independent Roman domination on graphs is an attractive topic in graph theory,and the definition is described as follows.Given a graph G=(V,E),a function f:V(G)→{0,1,2}is an outer-independent Roman dominating function(OIRDF)if f satisfies that every vertex v∈V with f(v)=0 has at least one adjacent vertex u∈N(v)with f(u)=2,where N(v)is the open neighborhood of v,and the set V0={v|f(v)=0}is an independent set.The weight of an OIRDF f is w(f)=∑_(v∈V)f(v).The value of minf w(f)is the outerindependent Roman domination number of G,denoted asγoiR(G).This paper is devoted to the study of the outer-independent Roman domination number of the Cartesian product of paths P_(n)□P_(m).With the help of computer,we find some recursive OIRDFs and then we present an upper bound ofγoiR(P_(n)□P_(m)).Furthermore,we prove the lower bound ofγoiR(P_(n)□P_(m))(n≤3)is equal to the upper bound.Hence,we achieve the exact value ofγoiR(P_(n)□P_(m))for n≤3 and the upper bound ofγoiR(P_(n)□P_(m))for n≥4.
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The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.
基金The National Natural Science Foundation of China(No.10971025)
文摘Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L（j, k）-labeling of G is called the circular L（j, k）-labeling number of G and is denoted by σj, k（G）. For any two positive integers j and k with j≤k,the circular L（j, k）-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined.
文摘A set ?is a dominating set of G if every vertex of ?is adjacent to at least one vertex of S. The cardinality of the smallest dominating set of G is called the domination number of G. The square G2 of a graph G is obtained from G by adding new edges between every two vertices having distance 2 in G. In this paper we study the domination number of square of graphs, find a bound for domination number of square of Cartesian product of cycles, and find the exact value for some of them.
文摘Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, ?(mod 3) and bounds for otherwise.
基金supported by the National Natural Science Foundation of China(Grant No.61966007)Key Laboratory of Cognitive Radio and Information Processing,Ministry of Education(No.CRKL180106,No.CRKL180201)+1 种基金Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing,Guilin University of Electronic Technology(No.GXKL06180107,No.GXKL06190117)Guangxi Colleges and Universities Key Laboratory of Satellite Navigation and Position Sensing.
文摘In this paper,we focus on inferring graph Laplacian matrix from the spatiotemporal signal which is defined as“time-vertex signal”.To realize this,we first represent the signals on a joint graph which is the Cartesian product graph of the time-and vertex-graphs.By assuming the signals follow a Gaussian prior distribution on the joint graph,a meaningful representation that promotes the smoothness property of the joint graph signal is derived.Furthermore,by decoupling the joint graph,the graph learning framework is formulated as a joint optimization problem which includes signal denoising,timeand vertex-graphs learning together.Specifically,two algorithms are proposed to solve the optimization problem,where the discrete second-order difference operator with reversed sign(DSODO)in the time domain is used as the time-graph Laplacian operator to recover the signal and infer a vertex-graph in the first algorithm,and the time-graph,as well as the vertex-graph,is estimated by the other algorithm.Experiments on both synthetic and real-world datasets demonstrate that the proposed algorithms can effectively infer meaningful time-and vertex-graphs from noisy and incomplete data.
文摘A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].
文摘In this paper we prove that the complete bipartite graph kmn where m and n are even, join of two cycle graphs cn and cm where n + m ≡ 0 (mod 4), split graph of cn for even “n”, Kn × P2 where n is even are admits a Zero-M-Cordial labeling. Further we prove that Kn × P2Bn = K1,n × P2 of odd n admits a Zero-M-Cordial labeling.
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Sponsored by the National Natural Science Foundation of China(Grant Nos.11361024,61472138)the Provincial Natural Science Foundation(Grant Nos.20171BAB201009,20161BAB202066)the Jiangxi Provincial Science and Technology Project(Grant No.KJLD12067)
文摘Let G =( V,E) be a connected graph and W = { w_1,w_2,…,w_k} be an ordered subset of V( G).For any vertex v ∈V,the locating code of v with respect to W is the k-vector CW( v) = { d( v,w_1),d( v,w_2),…,d( v,w_k) },W is said to be a locating set of G if distinct vertices have the distinct locating code,and the locating number of G is defined as: Loc( G) = min{ | W| : W is a locating set of G}.We study the locating set and locating number of a graph G,obtain some bounds for the locating numbers of graphs,and determine the exact value of Loc( G) for some special classes of graphs,such as cycles,wheels,complete t-partite graph and some Cartesian products of paths and cycles. In addition,we also prove that Loc( T) ≥Δ-1 holds for all trees T with maximum degree Δ,and shows a tree T with Loc( T) = Δ-1.
文摘Abstract Let U(R,S) denote the class of all (0,1) m×n matrices having row sum vector R and column sum vector S. The interchange graph G(R,S) is the graph where the vertices are the matrices in U(R,S) and two vertices representing two such matrices are adjacent provided they differ by an interchange. It is proved that G(R,(1,1,...,1)) is a generalized Cartesian product of some Johnson Scheme graphs. Furthermore, its connectivity, diameter and transitivity (vertex ,edge ) are also determined.
基金the National Natural Science Foundation of China （No. 10771062） the Fund for New Century Excellent Talents in University （No. NCET-07-0276）.
文摘In this paper, the crossing numbers of the Cartesian products of a specific 5-vertex graph with a star are given, and thus the result fills up the crossing number list of Cartesian products of all 5-vertex graphs with stars （presented by Marian Klesc）. In addition, we also give an up to date description of Cartesian products of 5-vertex graphs with stars, whose crossing numbers are known.
