THE CHARACTERISTIC POLYNOMIAL OF MULTI-ROOTED, DIRECTED TREES LAUREN EATON Mathematics 8 May 2008 Independent Research Project: Candidate for Departmental Honors Graduation: 17 May 2008 THE CHARACTERISTIC POLYNOMIAL OF MULTI-ROOED, DIRECTED TREES LAUREN EATON Abstract. We define the characteristic polynomial for single-rooted trees and begin with a theorem about this polynomial, derived from the known contrac- tion/deletion formula. We expand our scope to include multi-rooted, directed trees. Introducing the concept of a star, we prove two theorems which allow us to evaluate the polynomials of these trees in terms of the stars that comprise them. Finally, we derive and prove a general formula for the characteristic polynomial of multi-rooted, directed trees. Introduction Introduced by Gary Gordon and Elizabeth McMahon in [3] the greedoid char- acteristic polynomial is a generalization of the Tutte polynomial, which is a two- variable invariant which helps describe the structure of a single-rooted graph. Given a multi-rooted, directed tree, one must conduct an often tedious recursive calcula- tion in order to obtain the polynomial. In this paper, we work towards defining a general formula for the characteristic polynomial that requires minimal calculation and is entirely in terms of the vertices. This paper is organized as follows: In Section 1, we start with a focus on single- rooted trees and an adaptation of the characteristic polynomial. In Section 2, we look at the more complex multi-rooted, directed trees. We will find that when deriving the characteristic polynomial of these complicated trees, one can consider them in simplified terms. We will explore the concept of a sink and see how the polynomial is affected by its presence. Likewise, we introduce a graphical element called a star; a multi-rooted, directed tree can be thought of as being composed entirely of stars with shared edges. We show the polynomial 1 2 LAUREN EATON of this tree is equivalent to a product whose terms are polynomials of the stars that it is composed of. This leads to our final theorem, a formula for calculating the polynomial of a multi-rooted, directed tree in terms of the contributions of its vertices. 1. Single-Rooted Trees Definitions. In this paper, we expand upon published research conducted by Gary Gordon and Elizabeth McMahon [1], [2], [3] and thus reference their work through- out. We recall various definitions from graph theory, first those pertinent and relating to single-rooted trees and then those related to multi-rooted trees. Let G = (V, E, r) be a rooted graph. A graph G is a finite nonempty set of vertices together with a set of unordered pairs of distinct vertices of G called edges. Specifically, G is a single-rooted tree with edge set E(G), vertex set V (G), and root r, where a root is any distinguished vertex. A tree is defined as a graph in which any two vertices are connected by exactly one path. A path is a finite, alternating sequence of incident vertices and edges in which no vertex is repeated. A tree is a connected graph, because, given any u, v V , there exists a path between u and v ∈ in G. A tree is also an acyclic connected graph. An acyclic graph contains no cycle — a nontrivial closed finite, alternating sequence of vertices and edges in which no edge is repeated. The degree of a vertex v in G, denoted degG(v), is the number of edges incident with v. Define L to be the set of leaves — vertices, excluding roots, with degree equal to 1— L = v V r : deg (v) = 1 , and let ! = L . For { ∈ − G } | | further definitions, see [5]. Let G be a subgraph of G, denoted G G. A graph G is a subgraph of G if i i ⊆ i V (G ) V (G) and E(G ) E(G). Also, the root r is an element of G . i ⊆ i ⊆ i Definition 1. If G and G are subgraphs of G such that E(G ) E(G ) = , 1 2 1 2 ∅ V (G ) V (G ) = r , E(G ) E(G ) = E(G) and V (G ) V (G )!= V (G), then 1 2 { } 1 2 1 2 ! " " THE CHARACTERISTIC POLYNOMIAL OF MULTI-ROOED, DIRECTED TREES 3 G is the direct sum of G and G , and we denote G = G G . 1 2 1 ⊕ 2 We use E and E(G) interchangeably. Similarly, E = E(G ). Suppose S E. i i ⊆ Definition 2. Let rG(S) denotes the rank of S in G: the number of vertices con- nected to the root by some path in S. The rank does not include the root. Similarly, for G G and S = S E , let r (S ) correspond to the rank of S in G . i ⊆ i ∩ i Gi i i i Suppose G = G G ; then, E and E form a partition on E with r (E) = 1 ⊕ 2 1 2 G r (E )+r (E ). Define S = S E and S = S E , so S = S S and S E G1 1 G2 2 1 ∩ 1 2 ∩ 2 1 ∪ 2 1 ⊆ 1 and S E . It follows that r (S) = r (S ) + r (S ). Let S be defined as 2 ⊆ 2 G G1 1 G2 2 | | the number of elements in the set S. For conciseness we define the corank of S to be cor (S) = r (E) r (S) and cor (S ) = r (E ) r (S ). Likewise, we G G − G Gi i Gi i − Gi i define the nullity of S to be nul (S) = S r (S) and nul (S ) = S r (S ). G | | − G Gi i | | − Gi i As defined by Gary Gordon and Elizabeth McMahon, the characteristic polynomial p(G; λ) is expressed, in terms of the corank and nullity, by: Proposition 1 (characteristic polynomial; [1]). p(G; λ) = ( 1)rG(E) ( λ)corG(S)( 1)nulG(S) − − − S E #⊆ In the first theorem, we will demonstrate an alternate way of expressing the char- acteristic polynomial as identified above. Also defined by Gordon and McMahon is the Deletion-Contraction Formula for the characteristic polynomial and the Direct Sum Property, which we will make use of later. Let e be an edge incident to the root. The deletion of e means the erasing it from the graph to obtain new edge set E e and vertex set V r where r is the − − root from which e emanates. The contraction of e means erasing e and merging the two endpoints of e to obtain new edge set E e. Since e emanates from a root, we − merge the endpoints to form a new root. 4 LAUREN EATON Proposition 2 (Deletion-contraction; Prop. 3 [3]). r (E) r (E e) p(G; λ) = λ G − G − p(G e; λ) p(G/e; λ). − − The following is an example of the computation of a polynomial of a single-rooted tree using the deletion-contraction formula. Γ = ◦ • ◦ ? ?? ?? ?? ◦ ◦ e p(Γ; λ) = p • ◦ ?? ? ?? ?? ◦ ◦ ◦ = λ3 p p · − ◦ ?? • ?? ? ? ?? ?? ?? ?? ◦ ◦ = 0 ( 1)2(1 λ)2 − − − We will justify this step later By Direct Sum Property = ( 1)3(1 λ)*+,-2. * +, - − − Also, if G = G G , then: 1 ⊕ 2 Proposition 3 (Direct Sum Property; Prop. 4 [3] ). p(G G ) = p(G ) p(G ). 1 ⊕ 2 1 · 2 THE CHARACTERISTIC POLYNOMIAL OF MULTI-ROOED, DIRECTED TREES 5 We begin with the proof of our first theorem, a restating of the characteristic polynomial in Proposition 1. Theorem 1. The characteristic polynomial of a single-rooted tree G is given by E ! p(G; λ) = ( 1)| |(1 λ) . − − In order to prove Theorem 1, we begin by proving Proposition 3. Lemma 1. For G as defined above with G = G G , i 1 ⊕ 2 p(G; λ) = p(G ; λ) p(G ; λ). 1 · 2 G(S) G(S) corG (Si) nulG (Si) Proof. Let X(S) = ( λ)cor ( 1)nul and X (S ) = ( λ) i ( 1) i − − i i − − for i = 1, 2. In order to show X(S) = X (S ) X (S ), we will show cor (S) = 1 1 · 2 2 G corG1 (S1) + corG2 (S2) and nulG(S) = nulG1 (S1) + nulG2 (S2). First, consider cor (S ) + cor (S ) = r (E ) r (S ) + r (E ) r (S ) G1 1 G2 2 G1 1 − G1 1 G2 2 − G2 2 = r (E ) + r (E ) (r (S ) + r (S )) G1 1 G2 2 − G1 1 G2 2 = r (E) r (S) G − G = corG(S). Now, consider nul (S ) + nul (S ) = S r (S ) + S r (S ) G1 1 G2 2 | 1| − G1 1 | 2| − G2 2 = S + S (r (S ) + r (S )) | 1| | 2| − G1 1 G2 2 = S r (S) | | − G = nulG(S). 6 LAUREN EATON Then, X (S ) X (S ) simplifies as follows: 1 1 · 2 2 corG (S1) corG (S2) nulG (S1) nulG (S2) X (S ) X (S ) = ( λ) 1 ( λ) 2 ( 1) 1 ( 1) 2 1 1 · 2 2 − − − − corG (S1)+corG (S2) nulG (S1)+nulG (S2) = ( λ) 1 2 ( 1) 1 2 − − = ( λ)corG(S)( 1)nulG(S) − − = X(S). Finally, taking into account X (S ) X (S ) = X(S), examine p(G; λ) = X(S) 1 1 · 2 2 S E as follows: #⊆ p(G; λ) = X(S) S E #⊆ = X(S S ) 1 ∪ 2 S S E 1∪#2⊆ = X(S S ) 1 ∪ 2 S E S E 1#⊆ 1 2#⊆ 2 = X (S ) X (S ) 1 1 · 2 2 S E S E 1#⊆ 1 2#⊆ 2 = X (S ) X (S ) 1 1 2 2 S E S E 1#⊆ 1 2#⊆ 2 = p(G ; λ) p(G ;λ).