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A New Graph Invariant

Abstract The well-known Euler characteristic is an invariant of graphs defined by means of the vertex, edge and face numbers of a graph, to determine the genus of the underlying surface of the graph. By means of it, it is possible to determine the vertex, edge and face numbers of all possible graphs which can be drawn in a given orientable/non-orientable surface. In this paper, by means of a given degree sequence, a new number denoted by ( ) G Ω which is related to Euler characteristic and has several applications in Graph Theory is defined. This formula gives direct information compared with the Euler characteristic on the realizability, number of realizations, connectedness, being acyclic or cyclic, number of components, chords, loops, pendant edges, faces, bridges etc.


Introduction
As usual, let G = (V, E) be a graph with |V(G)|= n vertices and |E(G)|= m edges. n and m are called the order and the size of the graph G. For a vertex v ∈ V (G), the degree of v is denoted by d v . A vertex with degree one which is one of the fundamental concepts of these paper is called a pendant vertex. With slight abuse of language, we shall use the term "pendant edge" for an edge having a pendant vertex. The smallest and biggest vertex degrees in a graph will be denoted by δ and ∆, respectively. If u and v are adjacent vertices of G, and if the edge e connects them, this situation will be denoted by e = uv. In such a case, the vertices u and v are called adjacent vertices and the edge e is said to be incident with u and v. Adjacency and incidency play a very important role in the spectral graph theory, the sub-area of graph theory dealing with linear algebraic study of graphs.
Written with multiplicities, a degree sequence in general is written as Let D be a set of some non-decreasing non-negative integers. We say that a graph G is a realization of the set D if the degree sequence of G is equal to D. It is clear from the definition that for a realizable degree sequence, there is at least one graph having this degree sequence. For example, the completely different two graphs in Figure 1 have the same degree sequence.
There are many results and algorithms to determine whether a given set of non-negative integers is realizable or not, the most famous one being Havel-Hakimi's, see [1,2]. Of course, the most obvious criteria that arises from the handshaking lemma says that the sum of all vertex degrees, being equal to twice the number of edges, must be even.
We shall classify our graphs according to whether they have at least one cycle or not. Those graphs having no cycle will be called acyclic. For example, all trees are acyclic. The remaining graphs are called cyclic graphs.

How the Invariant Arose?
It is well-known that the number a 1 of leaves (pendant vertices) of any given tree T is given by ( ) Motivated by this fact, the authors studied a large number of graph realizations of given realizable degree sequences and observed that the left hand side of this equation is a graph invariant. We denote this number by Ω. This invariant has a big potential to give structural and combinatorial information about a graph just by means of its degree sequence: With slight abuse of language, we shall sometimes use the expression Ω(D) for the Ω of a given degree sequence. To understand Ω, let us take the graph G in Figure 2. The Ω(G) of G is ( ) 1 1 2 1 3 1 5 1 1 1

For any graph G,Ω (G) is even.
This has the following obvious test for the realizability of a given degree sequence: Corollary 2.1. Let D be a set of non-negative integers. If Ω(D) is odd, then D is not realizable.
As usual, let P n , C n , S n , K n , T r,s and K r,s , where n = r + s denote the path, cycle, star, complete, tadpole and complete bipartite graphs with n vertices, respectively. Let T denote a tree. For these well-known graph classes, the Ω(G) is as follows. One must note that the Ω of a path, star or tree is equal to −2. This is in fact true for all connected acyclic graphs as we shall see.

Properties of Ω
We now deal with some properties of Ω. We first study the number r of the closed regions (faces) which are bounded by the edges of the graph G. That is, the region outside the graph does not count. Note that a closed region could be bounded by any n-cycle (n-gon) where n ≥ 3, a loop (1-gon) or multiple edges (2-gon).  As we shall often deal with disconnected graphs, the following useful result that shows that Ω(G) of a graph G is additive over the set of the components of G will be needed: Theorem 3.2. Let G be a disconnected graph with c components 1 2 , ,..., .
The following is a direct generalization of Theorem 3.1 to disconnected graphs:     G Ω = .
The following is an interesting relation between Ω, m and n: We shall often come up with unicyclic graphs which are graphs with only one closed region, that is a cycle with at least three edges, a pair of multiple edges or a loop. By Theorem 3.1, we have Ω = 0 for all unicyclic graphs. The following result then directly follows: Corollary 3.5. Let G be a connected graph. If ( ) 0, Similarly, we can say that for bicyclic and three-cyclic connected graphs, we have the relations m = n + 1 and m = n + 2, respectively.
We also have Lemma 3.3. Let G be a graph with m > n = 3. Then G has at least one multiple edge or loop. Proof. If n = 3, then form a triangle using three of m > 3 edges and all three vertices. If we do not form a triangle, we have to have a loop or multiple edges. As there is no place for chords in a triangle, all the remaining edges must be loops or multiple edges.