A Two Parameter Chromatic Symmetric Function

We introduce and develop a two-parameter chromatic symmetric function for a simple graph G over the field of rational functions in q and t , Q (q, t). We derive its expansion in terms of the monomial symmetric functions, mλ, and present various correlation properties which exist between the two-parameter chromatic symmetric function and its corresponding graph. Additionally, for the complete graph G of order n, its corresponding two parameter chromatic symmetric function is the Macdonald polynomial Q(n). Using this, we develop graphical analogues for the expansion formulas of the two-row Macdonald polynomials and the two-row Jack symmetric functions. Finally, we introduce the “complement” of this new function and explore some of its properties. 1. Preliminaries. We briefly define some of the basic concepts used in the development of our two parameter chromatic symmetric function. In general, our notation will be consistent with that of [1]. Let G be a finite, simple graph; G has no multiple edges or loops. Denote the edge set of G by E(G) and the vertex set of G by V (G). The order of the graph G, denoted o(G), is the size of its vertex set V (G) and the size of the graph G, denoted s(G), is equal to the number of edges in E(G). A subgraph of G , G , is a graph whose vertex set and edge set are contained in those of G. For a subset V (G) ⊆ V (G), the subgraph induced by V (G) , GI , is the subgraph of G which contains all edges in E(G) which connect any two vertices in V (G). For the graph G, denote the edge of E(G) which joins the vertices vi, vj ∈ V (G) by vivj ; we say that vi and vj are the endvertices of the edge vivj. A walk in G is a sequence of vertices and edges, v1, v1v2, . . . , vl−1vl, vl, denoted v1 . . . vl; the length of this walk is l. A path is a walk with distinct vertices and a trail is a walk with distinct edges. A trail whose endvertices are equal, v1 = vl, is called a circuit. A walk of length ≥ 3 whose vertices are all distinct, except for coinciding endvertices, is called a cycle. The graph G the electronic journal of combinatorics 14 (2007), #R22 1 is said to be connected if for every pair of vertices {vi, vj} ∈ V (G), there is a path from vi to vj. A tree is a connected, acyclic graph. Let V (G) = {v1, . . . , vn}. Denote the number of edges emminating from the vertex vi ∈ V (G) by d(vi), the degree of the vertex vi. The degree sequence of G, denoted by deg(G), is a weakly decreasing sequence (or partition) of nonnegative integers, deg(G) = (d1, . . . , dn), such that the length of deg(G) is equal to |V (G)| and (d1, . . . , dn) represents the degrees of the vertices of V (G), arranged in decreasing order. Since each edge of G has two endvertices, it follows that ∑n i=1 di = 2s(G); thus, deg(G) ` 2s(G). A coloring of the graph G is a function k : V (G) → N. The coloring k is said to be proper if k(vi) 6= k(vj) whenever vivj ∈ E(G). Additionally, we will use the following consistent with [2]. (a; q)0 = 1 (a; q)n = n−1