A study on block related derived graphs and their signed graphs

Abstract

The derived graphs related to blocks are explored in this research. The notion of newlineblock-degree of the vertex and cut-degree of the block are introduced and studied its newlineproperties. This aids in exploring a few block-related derived graphs such as block newlinegraph of a graph and block cut point graph of a graph. The semi-splitting block graph SB (G) of a graph G is a graph derived from G with a vertex set consisting V and#8746; U and#8746; Band#8242;, where V = V (G), U is the set consisting of vertices ui which is added corresponding to each vertex vi in G and Band#8242; is the set consisting of vertices introduced for each block of G. Two vertices in SB (G) are adjacent if : (i) they are adjacent in newlineG, (ii) one corresponds to a vertex uj of U and the other corresponds to vi of V (G), newlinewhere vi is in NG (vj), (iii) one corresponds to bl of Band#8242; and other corresponds to vi newlineof V where vi is a vertex of the block Bl in G. The structural properties of the semisplitting block graph of a graph are studied. The parameters such as connectivity, newlinecovering, independence and chromatic number of the semi-splitting block graph of a graph are investigated exhaustively. In addition, the distance parameters such as radius, central vertices, diameter, and diametral path are analyzed. The triangle degree of a vertex tG(v) is the number of triangles in G containing that vertex. The triangle degree of the vertices of semi-splitting block graph of a graph is determined. The triangle degree signed graph St(G) of a graph G is obtained from G in which the edge uv receives the sign (and#8722;1)|tG(u)and#8722;tG(v)|. A T and#8722;marking is a marking on a signed graph where the vertices receive the sign (and#8722;1)tG(v). A few properties of St(SB(G)) based on the signed graph and marked (T and#8722;marking) signed graph are studied. The labeling called T and#8722;cordial and total T and#8722;cordial are introduced and certain classes of graphs which admit T and#8722;cordial and total T and#8722;cordial are studied.