www.iaset.us editor@iaset.us TOTAL PATHOS TOTAL VERTEX SEMIENTIRE BLOCK GRAPH VENKANAGOUDA M GOUDAR 1 , RAJANNA N E 2 & C K SUBBARAYA 3 1 Department of Mathematics, Sri Siddhartha Institute of Technology, Tumkur, Karnataka, India 2 Research scholar Sri Siddhartha Academy of Higher Education, Tumkur, Department of Mathematics, Adichunchanagiri Institute of Technology, Chikmagalur, Karnataka, India 3 Department of Computer Science and Engineering, Adichunchanagiri Institute of Technology, Chikmagalur, Karnataka, India ABSTRACT In this paper, we introduce the concept of total pathos total vertex semientire block graph of a tree. We obtain some properties of this graph. We study the characterization of graphs whose of total pathos total vertex semientire block graph of a tree is Hamiltonian, nonplanar and noneulerian. KEYWORDS: Block Graph, Inner Vertex Number, Line Graph, Path of Pathos, Vertex Semientire Graph Mathematics Subject Classification: 05C 1. INTRODUCTION All graphs considered here are finite, undirected without loops or multiple edges. Any undefined term or notation in this paper may be found in Harary [2]. The concept of pathos of a graph G was introduced by Harary [2], as a collection of minimum number of line disjoint open paths whose union is G. The path number of a graph G is the number of paths in pathos. For a graph G(p, q) if B = { u 1 , u 2 , u 3 , · · · , u r ; r ≥2} is a block of G , then we say that point u 1 and block B are incident with each other, as are u 2 and B and so on. If two distinct blocks B 1 and B 2 are incident with a common cut vertex then they are called adjacent blocks. By a plane graph G we mean embedded in the plane as opposed to a planar graph. In a plane graph G, let e 1 ={u,v }be an edge. We say e 1 is adjacent to the vertices u and v, which are also adjacent to each other. Also an edge e 1 is adjacent to the edge e 2 =uw. A region of G is adjacent to the vertices and edges which are on its boundary, and two regions of G are adjacent if their boundaries share a common vertex. The edgedegree of an edge e = {a, b} is the sum of degrees of the end vertices a and b. Degree of a block is the number of vertices lies on a block. Blockdegree B v of a vertex v is the number of blocks in which v lies. Degree of a region is the number of vertices lies on a region. The regiondegree R v of a vertex v is the number of regions in which the vertex v lies. A new concept of a graph valued functions called the vertex semientire block graph e vb (G) of a plane graph G was introduced by Venkanagouda in [9] and is defined as the graph whose vertex set is the union of the set of vertices, blocks International Journal of Applied Mathematics & Statistical Sciences (IJAMSS) ISSN(P): 2319-3972; ISSN(E): 2319-3980 Vol. 5, Issue 3, Apr - May 2016; 53-62 © IASET