Power domination in splitting and degree splitting graph

A vertex set S is called a power dominating set of a graph G if every vertex within the system is monitored by the set S following a collection of rules for power grid monitoring. The power domination number of G is the order of a minimal power dominating set of G. In this paper, we solve the power domination number for splitting and degree splitting graph.

Prime labeling of line and splitting graph of brush graph

A bijective function f from V(G) to {1,2,…, n} be a prime labeling of a graph G with n order if for every u, v ∈ V(G) such that e = uv ∈ E(G), f(u) and f(v) relatively prime. A prime graph is a graph which admits prime labeling. In this study, we investigate and conclude that the line and splitting graph of the brush graph is a prime graph.

Broader families of cordial graphs

<p>A binary labeling of the vertices of a graph <em>G</em> is cordial if the number of vertices labeled 0 and the number of vertices labeled 1 differ by at most 1, and the number of edges of weight 0 and the number of edges of weight 1 differ by at most 1. In this paper we present general results involving the cordiality of graphs that results of some well-known operations such as the join, the corona, the one-point union, the splitting graph, and the super subdivision. In addition we show a family of cordial circulant graphs.</p>

ON STAR COLORING OF DEGREE SPLITTING OF COMB PRODUCT GRAPHS

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number χs (G) of G is the least number of colors needed to star color G. Let G = (V,E) be a graph with V = S1 [ S2 [ S3 [ . . . [ St [ T where each Si is a set of all vertices of the same degree with at least two elements and T =V (G) − St i=1 Si. The degree splitting graph DS (G) is obtained by adding vertices w1,w2, . . .wt and joining wi to each vertex of Si for 1 i t. The comb product between two graphs G and H, denoted by G ⊲ H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the ith copy of H at the vertex o to the ith vertex of G. In this paper, we give the exact value of star chromatic number of degree splitting of comb product of complete graph with complete graph, complete graph with path, complete graph with cycle, complete graph with star graph, cycle with complete graph, path with complete graph and cycle with path graph.

Splitting on classes of helm graphs and its equitabletotal chromatic number

The equitable total coloring of a graph $G$ is the different colors used to color all the vertices and edges of $G$, in the order that adjacent vertices and edges are assigned with least different $k$-colors and can be partitioned into colors sets which differ by maximum one. The minimum of $k$-colors required is known as the equitable total chromatic number. In this paper the splitting graph of Helm and Closed Helm graph is constructed and its equitable total chromatic number is acquired.

BILANGAN DOMINASI TOTAL PADA TRIANGULAR SNAKE GRAPH

Graf dengan adalah himpunan titik dan adalah himpunan sisi yang menghubungkan sepasang titik. Himpunan , disebut himpunan dominasi pada graf jika semua titik yang tidak berada pada himpunan bertetangga sedikitnya dengan satu titik dari dan kardinalitas minimum dari disebut bilangan dominasi . Persekitaran atau adalah himpunan semua titik di yang bertetangga dengan titik-titik di Himpunan adalah himpunan dominasi total dalam jika dan kardinalitas minimum dari himpunan dominasi total dari graf dilambangkan dengan . Penelitian ini membahas tentang bilangan dominasi total pada triangular snake graph , line graph dan splitting graph dari triangular snake graph . Triangular snake graph adalah suatu bentuk graf yang diperoleh dari lintasan graf dengan mengganti semua sisi lintasan graf dengan segitiga Line graph adalah bentuk pemberian titik pada setiap sisi yang bersisian di graf . Sedangkan splitting graph adalah graf yang diperoleh dengan menambahkan titik baru sesuai dengan titik dari sehingga Hasil dari penelitian diperoleh bilangan dominasi total pada triangular snake graph, line graph dan splitting graph dari triangular snake graph yaitu , dan Kata Kunci: persekitaran, line graph, splitting graph

Divisor Degree Energy of Some Graphs using Mat Lab Programme

In this paper, we find the divisor degree matrix of wheel, fan, complete bipartite and splitting graph of star graph and generate their divisor degree energy. Also, we obtain the divisor degree matrix of a tree graphs such as path, star and comb graph and their divisor degree energy by using MAT LAB programme.