Doctoral Theses

Some External Problems and Charecterization in the Theory of Graphs.

2-28-1969

2-28-1970

Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

Degree Name

Doctor of Philosophy

Mathematics

Department

Research and Training School (RTS)

Abstract (Summary of the Work)

Graph theory has become such a well known and widely applied subject that it is not nocessary to give a general Introduetion to it. Instead ve eive helov a summary of the results of the thesis chapterwise.The study of extremal problems in Graph theory was started by Turan who deternined the minimum independence number of a graph on n vertices vith m edges. More recently Harary determined the maxinum conneetivity of a graph on n vortices vith m edges.In a paper concerning the degrees of the vertices of a graphy laktnd posed the following two problems : dotermin the maximum number of cut vertices (cut odges) in a graph on n vertices with given degrees d, d *** d, We solve some rolated problens in Chapter l.A cut vertex (cut edge) of a graph is a vertex (edge) vhose removal increases the number of components of the graph In Chapter l, we consider the deteraination of the range of the number of cut vertices (cut edges) in a craph on vertices with n edges under cach of the folloving two conditions.(1) The degree of no vortox is less than d.(2) the degree of no vertex is groater thnn de problem is solved completely for d < 3 and when da 4 and Tho extremal graphs are characterised when ds 2. The range of the nunber of beidges in a Inatro1d with m eells, rank ny and containing no atreuit of length loss than k Is al so determinede.The power of a vortex x of a connected eraph G may bo dafined as tho nuber of components of G-x. lecessary and sufficiont conditions for positive integors pPa..Pn to bo the powors of the vertices of some graph on n vertices (also vith n edges) are g1ven in Chapter 2. The naxinun. power of a vertex of a connoe ted eraph vith n vertions and n edges is also dotormined. Necessary and suf iciont conditions for the axistence of an (undirectod) greph with dogrees dg, de results of liale, Ryaer and Fulkerson for directud graphs. The sane problm is slso solved under each of the conditions (1) the graph is connected (11) the graph is biconnacted. Thene results have been abtained earlicr, using different ... 4nra deduced from the corresponding methods by Brdos and Gallal, Ilakini and Tutte.The line graph of a graph G may be do fined ns the graph whoso vorticer are tho odges of Gr two vertices or L(G) being adjacont if and only if the corrosponding edgos of G are adjacent. Some sinple properties and characterita- tions of line graphs are clven in Chapter 3. In particular, a characterization of the 1ine graph of the complete A-partite eraph Kun..n to givon. Tiis Ceneralitos the reslts of Connor and Shrikhando on the line gtaphn of the triangular and Igannoa tation mehimos.In Chapter 4, wo obtzin a characterization of the graph GlApn) definod as follows : tako k disjolnt sets 5...3 2ach with n olononta nhd lot , 8.

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