A tree T is felicitous if there is a labelling l of its vertices with distinct integers from the set {0,1,2,…,｜E(T)｜}, so that the induced edge labelling l′ defined by l′(e)=l(u)+l(v) mod ｜E(T)｜ for eac...A tree T is felicitous if there is a labelling l of its vertices with distinct integers from the set {0,1,2,…,｜E(T)｜}, so that the induced edge labelling l′ defined by l′(e)=l(u)+l(v) mod ｜E(T)｜ for each e=uv∈E(T), assigns each edge e a different label. In this paper, we constructively proved that more classes of trees are felicitous. In the end, we gave a conjecture that every lobster tree is felicitous.展开更多
Let G=Cay^(+)(Z_(p1)×Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk),Φ)be a simple graph having vertex set V(G)=Z_(p1)×Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk)and edge set E(G)={{x,y}:x+y∈Φ},where all p_(1),p_(2),…,p_(...Let G=Cay^(+)(Z_(p1)×Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk),Φ)be a simple graph having vertex set V(G)=Z_(p1)×Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk)and edge set E(G)={{x,y}:x+y∈Φ},where all p_(1),p_(2),…,p_(k)are distinct prime factors andΦis the set of all units of the ring Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk).LetΣ=(G,σ)be a signed graph whose underlying graph is G and signature function isσ:E(G)→{+1,-1}defined asσ({x,y})={+1,ifx∈φ(p_(1))or y∈φ(p_(1)^(α1)p_(2)^(α2)…p_(k)^(αk));-1,otherwise.In this paper,we characterize the balance ofΣand some graphs derived from it such asη(Σ),L(Σ)and C_(E)(Σ).Moreover,we investigate the clusterability and sign-compativility ofΣ.展开更多
文摘A tree T is felicitous if there is a labelling l of its vertices with distinct integers from the set {0,1,2,…,｜E(T)｜}, so that the induced edge labelling l′ defined by l′(e)=l(u)+l(v) mod ｜E(T)｜ for each e=uv∈E(T), assigns each edge e a different label. In this paper, we constructively proved that more classes of trees are felicitous. In the end, we gave a conjecture that every lobster tree is felicitous.
基金Supported by the National Natural Science Foundation of China(11561042,11961040)the Natural Science Foundation of Gansu Province(20JR5RA 418)
文摘Let G=Cay^(+)(Z_(p1)×Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk),Φ)be a simple graph having vertex set V(G)=Z_(p1)×Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk)and edge set E(G)={{x,y}:x+y∈Φ},where all p_(1),p_(2),…,p_(k)are distinct prime factors andΦis the set of all units of the ring Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk).LetΣ=(G,σ)be a signed graph whose underlying graph is G and signature function isσ:E(G)→{+1,-1}defined asσ({x,y})={+1,ifx∈φ(p_(1))or y∈φ(p_(1)^(α1)p_(2)^(α2)…p_(k)^(αk));-1,otherwise.In this paper,we characterize the balance ofΣand some graphs derived from it such asη(Σ),L(Σ)and C_(E)(Σ).Moreover,we investigate the clusterability and sign-compativility ofΣ.
