Let G = （V, E） be a graph, and let f ： V →{-1, 1} be a two-valued function. If ∑x∈N（v） f（x） ≥ 1 for each v ∈ V, where N（v） is the open neighborhood of v, then f is a signed total dominating function on ...Let G = （V, E） be a graph, and let f ： V →{-1, 1} be a two-valued function. If ∑x∈N（v） f（x） ≥ 1 for each v ∈ V, where N（v） is the open neighborhood of v, then f is a signed total dominating function on G. A set {fl, f2,… fd} of signed d total dominating functions on G with the property that ∑i=1^d fi（x） ≤ 1 for each x ∈ V, is called a signed total dominating family （of functions） on G. The maximum number of functions in a signed total dominating family on G is the signed total domatic number on G, denoted by dt^s（G）. The properties of the signed total domatic number dt^s（G） are studied in this paper. In particular, we give the sharp bounds of the signed total domatic number of regular graphs, complete bipartite graphs and complete graphs.展开更多
A pioneer ecological research was carried out to assess the credence of Domat AI-Jandal Lake at AI-JoufProvince in KSA for natural habitats, aquatic lives, agricultural and tourism resorts purposes. The results showed...A pioneer ecological research was carried out to assess the credence of Domat AI-Jandal Lake at AI-JoufProvince in KSA for natural habitats, aquatic lives, agricultural and tourism resorts purposes. The results showed a poor inhabitant of aquatic birds i.e. yellow wagtail （Motacillaflava）; Black-winged Stilt （Himantopus himantopus）; shelduck （Tadorna tadorna）; Squacco heron （Ardeola ralloides）; Moorhen （Gallinula chloropus）; the bolti （Tilapia nilti）. Equally, only four plant species were predominant in the lake area i.e. synonym （Tamarix aucheriana）, a large perennial grass Phragmites australis, Athel pine or tamrisk （Tamarix aphylla） and synonym Zygophyllum simplex. Twelve fishes failed to survive longer than 45 minutes while another 6 fishes survived only in tributary water. Student T-Test biostatics analysis regarding the concentrations of the Total Dissolved Salts （TDS） in designated three parts of the lake showed significantly higher （P〈0.01〈0.0001） than acceptable limits for human usage, aquatic and natural habitat and perhaps for agricultural purposes. Similarly, the pHs of the waters were significantly alkaline （pH 8.9） and Eutrification of the lake waters clearly stand inappropriate for irrigation unless treated. The domestic pollutants were scattered almost all over the beaches as well as within the lake itself denoting unsafe of the lake as neither for natural habitats nor for tourism unless further attention is given by the municipal of Al-Jouf展开更多
Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f ： V（G） → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f1, f2,..., fd} of disti...Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f ： V（G） → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f1, f2,..., fd} of distinct Roman k-dominating functions on G with the property that ∑di=1 fi（v） ≤ 2 for each v C V（G）, is called a Roman k-dominating family （of functions） on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by dkR（G）. Note that the Roman 1-domatic number dlR（G） is the usual Roman domatic number dR（G）. In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for dkR（G）. In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.展开更多
基金Project supported by the National Natural Science Foundation of China （Grant No.1057117）, and the Science Foundation of Shanghai Municipal Commission of Education （Grant No.05AZ04）.
文摘Let G = （V, E） be a graph, and let f ： V →{-1, 1} be a two-valued function. If ∑x∈N（v） f（x） ≥ 1 for each v ∈ V, where N（v） is the open neighborhood of v, then f is a signed total dominating function on G. A set {fl, f2,… fd} of signed d total dominating functions on G with the property that ∑i=1^d fi（x） ≤ 1 for each x ∈ V, is called a signed total dominating family （of functions） on G. The maximum number of functions in a signed total dominating family on G is the signed total domatic number on G, denoted by dt^s（G）. The properties of the signed total domatic number dt^s（G） are studied in this paper. In particular, we give the sharp bounds of the signed total domatic number of regular graphs, complete bipartite graphs and complete graphs.
文摘A pioneer ecological research was carried out to assess the credence of Domat AI-Jandal Lake at AI-JoufProvince in KSA for natural habitats, aquatic lives, agricultural and tourism resorts purposes. The results showed a poor inhabitant of aquatic birds i.e. yellow wagtail （Motacillaflava）; Black-winged Stilt （Himantopus himantopus）; shelduck （Tadorna tadorna）; Squacco heron （Ardeola ralloides）; Moorhen （Gallinula chloropus）; the bolti （Tilapia nilti）. Equally, only four plant species were predominant in the lake area i.e. synonym （Tamarix aucheriana）, a large perennial grass Phragmites australis, Athel pine or tamrisk （Tamarix aphylla） and synonym Zygophyllum simplex. Twelve fishes failed to survive longer than 45 minutes while another 6 fishes survived only in tributary water. Student T-Test biostatics analysis regarding the concentrations of the Total Dissolved Salts （TDS） in designated three parts of the lake showed significantly higher （P〈0.01〈0.0001） than acceptable limits for human usage, aquatic and natural habitat and perhaps for agricultural purposes. Similarly, the pHs of the waters were significantly alkaline （pH 8.9） and Eutrification of the lake waters clearly stand inappropriate for irrigation unless treated. The domestic pollutants were scattered almost all over the beaches as well as within the lake itself denoting unsafe of the lake as neither for natural habitats nor for tourism unless further attention is given by the municipal of Al-Jouf
文摘Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f ： V（G） → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f1, f2,..., fd} of distinct Roman k-dominating functions on G with the property that ∑di=1 fi（v） ≤ 2 for each v C V（G）, is called a Roman k-dominating family （of functions） on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by dkR（G）. Note that the Roman 1-domatic number dlR（G） is the usual Roman domatic number dR（G）. In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for dkR（G）. In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.
