On the boundary polynomial of a graph

TL;DR

This paper introduces the boundary polynomial of a graph, explores its algebraic properties, and demonstrates how it encodes various graph parameters and characterizes different graph classes.

Contribution

It defines the boundary polynomial, investigates its properties, and establishes relationships with graph parameters and classes, providing new algebraic tools for graph analysis.

Findings

Boundary polynomial encodes domination and connectivity parameters.

Graph classes are characterized by their boundary polynomial.

Relationships between boundary polynomials of related graphs are established.

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.