[Paper Review] Tricyclic graphs with exactly two main eigenvalues

Xiaoxia Fan, Yanfeng Luo|arXiv (Cornell University)|Dec 5, 2010

Graph theory and applications8 references4 citations

TL;DR

This paper completely characterizes all connected tricyclic graphs with exactly two main eigenvalues by leveraging the 2-walk (a,b)-linear property, which is equivalent to having exactly two main eigenvalues. The authors identify 30 specific graphs $H_i$ (for $i=1$ to $30$) and eight families ${\mathcal{G}}_j$ ($j=1$ to $8$) that exhaustively classify all such graphs, providing explicit parameterized families and verifying their 2-walk linearity via degree and sum conditions on vertices.

ABSTRACT

An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.

Motivation & Objective

• To determine all connected tricyclic graphs that have exactly two main eigenvalues, extending prior work on unicyclic and bicyclic graphs.
• To characterize these graphs using the 2-walk (a,b)-linear property, which is equivalent to having exactly two main eigenvalues.
• To classify all such graphs by identifying their base structures and the possible ways trees can be attached to them while preserving the two-main-eigenvalue property.
• To provide a complete and explicit list of all such graphs, including 30 specific graphs $H_i$ and eight parameterized families ${\mathcal{G}}_j$.

Proposed method

• Use the equivalence between graphs with exactly two main eigenvalues and 2-walk (a,b)-linear graphs, as established by Hagos (2002).
• Apply the condition $S(v) = a d(v) + b$ for all vertices $v$, where $S(v)$ is the sum of degrees of neighbors of $v$, and $a,b$ are rational constants.
• Analyze the base tricyclic subgraph $G_B$ of each graph $G$, which contains no pendant vertices and is one of eight possible types as classified in prior work.
• Use structural constraints from Lemmas 2.4 and 2.5 to limit path lengths and degree patterns in internal paths and cycles.
• Apply the formula $a = \frac{S(v)-S(u)}{d(v)-d(u)}$, $b = \frac{d(u)S(v) - d(v)S(u)}{d(v)-d(u)}$ for vertices of unequal degree to derive integer values of $a$ and $b$.
• Systematically enumerate all possible configurations of pendant vertices and degree sequences on base graphs, verifying 2-walk linearity and integrality of $a,b$.

Experimental results

Research questions

• RQ1 Which connected tricyclic graphs have exactly two main eigenvalues, and how can they be completely classified?
• RQ2 What structural constraints must a tricyclic graph satisfy to be 2-walk (a,b)-linear for rational $a,b$?
• RQ3 How do the degrees of vertices in the base tricyclic subgraph and their neighbor sums constrain the existence of such graphs?
• RQ4 What are the complete families of graphs (including parameterized families) that satisfy the two-main-eigenvalue condition?
• RQ5 Can all such graphs be generated from a finite set of base structures by attaching trees under specific degree and sum conditions?

Key findings

• All connected tricyclic graphs with exactly two main eigenvalues are completely classified into 30 specific graphs $H_i$ ($i=1$ to $30$) and eight families ${\mathcal{G}}_j$ ($j=1$ to $8$).
• The graphs $H_{24}$ to $H_{29}$ are 2-walk $(2,2)$-, $(1,4)$-, $(0,6)$-, $(3,-1)$-, $(2,1)$-, and $(1,3)$-linear respectively, with explicit degree and sum conditions verified.
• The graph $H_{30}$ is 2-walk $(2,2)$-linear and arises when three adjacent vertices in the base have degree 3 and the rest of the structure satisfies $a+b=4$.
• The family ${\mathcal{G}}_7$ consists of graphs with $a=3$, $b\geq1$, and is 2-walk $(3,b)$-linear, with $p=l=q=2$ required for consistency.
• For graphs in ${\mathcal{G}}_8$, the base is a triangle with pendant paths, and the condition $a=2$, $b=2$ holds, with $a+b=4$ and $S(v)=2d(v)+2$ for all vertices.
• No such graphs exist with $a$ or $b$ non-integer, and all valid configurations are exhausted by the 30 $H_i$ graphs and the eight ${\mathcal{G}}_j$ families.