An inequality for the number of vertices with an interval spectrum in edge labelings of regular graphs

TL;DR

This paper establishes an upper bound on the number of vertices with interval spectra in edge labelings of regular graphs, linking graph structure and labeling properties.

Contribution

It introduces a novel inequality relating the count of vertices with interval spectra to graph parameters in regular graphs.

Findings

Proves an inequality bounding vertices with interval spectra in regular graphs.

Connects the number of such vertices to graph's connected components and order.

Provides a theoretical framework for analyzing spectrum properties in graph labelings.

Abstract

We consider undirected simple finite graphs. The sets of vertices and edges of a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. For a graph $G$, we denote by $\delta(G)$ and $\eta(G)$ the least degree of a vertex of $G$ and the number of connected components of $G$, respectively. For a graph $G$ and an arbitrary subset $V_0\subseteq V(G)$ $G[V_0]$ denotes the subgraph of the graph $G$ induced by the subset $V_0$ of its vertices. An arbitrary nonempty finite subset of consecutive integers is called an interval. A function $\varphi:E(G)\rightarrow \{1,2,\dots,|E(G)|\}$ is called an edge labeling of the graph $G$, if for arbitrary different edges $e'\in E(G)$ and $e''\in E(G)$, the inequality $\varphi(e')\neq \varphi(e'')$ holds. If $G$ is a graph, $x$ is its arbitrary vertex, and $\varphi$ is its arbitrary edge labeling, then the set $S_G(x,\varphi)\equiv\{\varphi(e)/ e\in E(G), e \textrm{is incident with} x$\} is called a spectrum of the vertex $x$ of the graph $G$ at its edge labeling $\varphi$. If $G$ is a graph and $\varphi$ is its arbitrary edge labeling, then $V_{int}(G,\varphi)\equiv\{x\in V(G)/\;S_G(x,\varphi)\textrm{is an interval}\}$. For an arbitrary $r$-regular graph $G$ with $r\geq2$ and its arbitrary edge labeling $\varphi$, the inequality $$ |V_{int}(G,\varphi)|\leq\bigg\lfloor\frac{3\cdot|V(G)|-2\cdot\eta(G[V_{int}(G,\varphi)])}{4}\bigg\rfloor. $$ is proved.