## Doctoral Theses

### On the Geometrisability of Some Strongly Regular Graphs Related to Polar Spaces.

2-28-1997

2-28-1998

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)

#### Abstract (Summary of the Work)

GraphsA graph G = (VE) consists of a finite set V and a subset E of ). (Here () denotes the set of all 2-subsets of V.) Elements of Vare called the vertices and the elements of E are called the edges of the graph. So Vis the vertex set and E is the edge set of the graph G. Two vertices a, y are said to be adjacent if the pair {a, y} is an edge; otherwise they are non-adjacent. If two vertices are adjacent then each is called a neighbour of the other vertex.Sometimes the edges of a graph are ordered pairs of vertices and in this case the graph is called a directed graph. If both ends of an edge terminate at the same vertex then the edge is said to be a ;. If two edges of a graph join the same two vertices then the graph is said to have multiple edges. But in this thesis the graphs we consider are undirected, loop free and with no multiple edges.A graph G - (V. ) is said to be a subgraph of a graph G = (VE) if vc v and C E. If F = En then G is said to be an induccd subyraph of G. It is also called the induced subgraph on v. The complement of a graph G = (VE) is the graph G = (V () \\ E).An isomorphism of a graph G onto a graph G is a one to one correspondence between the vertices in G and the vertices in G such that a pair of vertices are adjacent in G iff the corresponding pair of vertices are adjacent in G. Two graphs are said to be isomorphic if there exists an isomorphism between them. In this thesis we identify two graphs if they are isomorphic. An isomorphism of a graph to itself is called an automorphism, The automorphisms of a graph form a group under composition; it is called the automorphism group of the graph.The degree of a vertex a in a graph G is the number of vertices in G which are adjacent to . A graph is regular of degree k if all its vertices are of degree k.The unique connected regular graph of degree two on n vertices is called the n-cycle. A graph on n vertices in which cach vertex is adjacent to all other vertices is called the complete graph on n vertices, denoted by K. A graph whose edge set is empty is called a null graph. In other words, a mull graph is the complement of a complete graph. If n,., ng are positive integers, the complete multi-partite graph Km has its vertex set partitioned into k sets of size n, ., T4 such that two vertices are adjacent iff they belong to different parts. The line graph L(G) of a graph G has the edges of G as its vertices; two vertices of L(G) are adjacent iff the corresponding edges of G intersect. The triangular graph T, is by definition the line graph of K. The line graph of the complete bipartite graph K is called the n xn grid.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842996

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