On the boundary polynomial of a graph

TL;DR

The paper introduces the boundary polynomial $B(G;x,y)$, a two-variable generating function over subsets of vertices based on the outer boundary, and links it to known graph invariants through algebraic identities and evaluations. It establishes core properties, such as $B(G;x,y)$ encoding $n$, $m$, component structure, and invariant relationships like $B(G;1,y)=(1+y)^n$ and the isolation criterion via a factor $(y+1)$, while connecting to $\gamma(G)$, $\gamma_R(G)$, and $k_v(G)$ via coefficient considerations. It then computes $B(G;x,y)$ for key graph families, develops transformation formulas under graph operations (join, corona, vertex addition, edge operations, subdivision), and shows that several classical graph classes are characterized by their boundary polynomials. The work also discusses limitations, giving examples of non-uniqueness where different graphs share the same boundary polynomial, highlighting both the descriptive power and boundaries of this new polynomial as a graph invariant.

Abstract

In this work, we introduce the boundary polynomial of a graph $G$ as the ordinary generating function in two variables $B(G;x,y):= \displaystyle\sum_{S\subseteq V(G)} x^{|B(S)|}y^{|S|}$, where $B(S)$ denotes the outer boundary of $S$. We investigate this graph polynomial obtaining some algebraic properties of the polynomial. We found that some parameters of $G$ are algebraically encoded in $B(G;x,y)$, \emph{e.g.}, domination number, Roman domination number, vertex connectivity, and differential of the graph $G$. Furthermore, we compute the boundary polynomial for some classic families of graphs. We also establish some relationships between $B(G;x,y)$ and $B(G^\prime;x,y)$ for the graphs $G^\prime$ obtained by removing, adding, and subdividing an edge from $G$. In addition, we prove that a graph $G$ has an isolated vertex if and only if its boundary polynomial has a factor ($y+1$). Finally, we show that the classes of complete, complete without one edge, empty, path, cycle, wheel, star, double-star graphs, and many others are characterized by the boundary polynomial.

Figures (1)

• Figure 1: Non-isomorphic graphs with same order, size, degree sequence and boundary polynomial.

Theorems & Definitions (37)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Corollary 3.1