The effect of edge and vertex deletion on omega invariant

Abstract

Recently the first and last authors defined a new graph characteristic called omega related to Euler characteristic to determine several topological and combinatorial properties of a given graph. This new characteristic is defined in terms of a given degree sequence as a graph invariant and gives a lot of information on the realizability, number of realizations, connectedness, cyclicness, number of components, chords, loops, pendant edges, faces, bridges etc. of the family of realizations. In this paper, the effect of the deletion of vertices and edges from a graph on omega invariant is studied.

Omega Index of Line and Total Graphs

“…It is shown that Ω(G) � 2(m − n) and therefore it is always an even number. It is shown that the omega characteristic gives us very powerful information about cyclicness and connectedness of all the realizations of a given degree sequence (see, e.g., [9]). In brief, it is shown that all realizations of a degree sequence D with Ω(D) ≤ − 4 must be disconnected; each connected realization of a degree sequence D with Ω(D) � −2 must be a tree; each connected realization of a degree sequence D with Ω(D) � 0 must be a unicyclic graph; each connected realization of a degree sequence D with Ω(D) � 2 must be a bicyclic graph, etc.…”

“…where c(G) is the number of components of G. For more properties of the omega index, see [10,11]. e effect of edge and vertex deletion on the omega index is studied in [9]. Next, we obtain the number of pendant vertices of a caterpillar tree which consists of a main path so that all vertices are having maximum distance 1 from the path.…”

Üçgensel sayılar ve graflar

“…For convenience, the omega invariant of a realization G of D is also denoted by Ω( ). Some properties of Ω can be found in [1,2,3,4,6]. We recall some very important properties of it here.…”

“…A degree sequence is = {1 ( 1 ) , 2 ( 2 ) , 3 ( 3 ) , . .…”

Fibonacci Graphs

“…Let G be a realization of a degree sequence D. We recall the definition and some properties of Ω(G) from [5,[7][8][9][10]. The number a 1 of leaves of a tree T is…”