Unicyclic graphs with strong reciprocal eigenvalue property
Volume
1081-3810
Unicyclic graphs with strong reciprocal eigenvalue property
Sasmita Barik
Bhaba Kumar Sarma
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S. BARIK†, M. NATH†, S. PATI† , AND B. K. SARMA†
AMS subject classifications. 15A18, 05C50.
1. Introduction. Let G be a simple graph on vertices 1, 2, . . . , n. The adjacency
matrix of G is defined as the n × n matrix A(G), with (i, j)th entry 1 if {i, j} is an
edge and 0 otherwise. Since A(G) is a real symmetric matrix,all its eigenvalues are
real. Throughout the spectrum of G is defined as
σ(G) = (λ1(G), λ2(G), · · · , λn(G)),
where λ1(G) ≤ λ2(G) ≤ · · · ≤ λn(G) are the eigenvalues of A(G). The largest
eigenvalue of A(G) is called the spectral radius of G and is denoted by ρ(G). If G is
connected then A(G) is irreducible,thus by the Perron-Frobenius theory, ρ(G) is
simple and is afforded by a positive eigenvector,called the Perron vector. A graph G is
said to be singular (resp. nonsingular) if A(G) is singular (resp. nonsingular).
Definition 1.1. [
7
] Let G1 and G2 be two graphs on disjoint sets of n and
m vertices,respectively. The corona G1 ◦ G2 of G1 and G2 is defined as the graph
obtained by taking one copy of G1 and n copies of G2, and then joining the i-th vertex
of G1 to every vertex in the i-th copy of G2.
∗Received by the editors April 6, 2006. Accepted for publication March 16, 2008. Handling
Editor: Bryan L. Shader.
†Department of Mathematics, IIT Guwahati, Guwahati-781039, India (,
, , ).
140
Note that the corona G1 ◦ G2 has n(m + 1) vertices and |E(G1)| + n(|E(G2)| + m)
edges. Let us denote the cycle on n vertices by Cn and the complete graph on n
vertices by Kn. The coronas C3 ◦ K2 and K2 ◦ C3 are shown in Figure 1.1.
G1
G2
G1 ◦ G2
G2 ◦ G1
Connected graphs in which the number of edges equals the number of vertices are
called unicyclic graphs [
7
]. The unique cycle in a unicyclic graph G is denoted by Γ.
If u is a vertex of the unicyclic graph G then a component T of G − u not containing
any vertex of Γ is called a tree-branch at u. We say that the tree-branch is odd (even)
if the order of the tree-branch is odd (even). Recall that the girth of a unicyclic graph
G is the length of Γ.
It is well known (see [
4
] Theorem 3.11 for example) that a graph G is bipartite if
and only if the negative of each eigenvalue of G is also an eigenvalue of G. In contrast
to the plus-minus pairs of eigenvalues of bipartite graphs,Barik,Pati and Sarma,
[
1
],have introduced the notion of graphs with property (R). Such graphs G have the
property that λ1 is an eigenvalue of G whenever λ is an eigenvalue of G. When each
eigenvalue λ of G and its reciprocal have the same multiplicity,then G is said to
have property (SR). It has been proved in [
1
] that when G = G1 ◦ K1,where G1 is
bipartite,then G has property (SR) (see Theorem 1.2). However,there are graphs
with property (SR) which are not corona graphs. For example,one can easily verify
that the graphs in Figure 1.2 have property (SR). Note that,in view of Lemma 2.3,
H1 cannot be a corona and that H2 is not a corona has been argued in [
1
]. Note also
that the graph H1 is unicyclic and H2 is not even bipartite.
H1
H2
The following result gives a class of graphs with property (SR).
Theorem 1.2. [
1
] Let G = G1 ◦ K1, where G1 is any graph. Then λ is an
eigenvalue of G if and only if −1/λ is an eigenvalue of G. Further, if G1 is bipartite
then G has property (SR).
In [
1
],it is proved that a tree T has property (SR) if and only if T = T1 ◦ K1, for
some tree T1. This was strengthened in [
2
] where it is proved that a tree has property
(R) if an only if it has property (SR).
In view of the previous example it is clear that a unicyclic graph with property
(SR) may not be a corona and this motivates us to study unicyclic graphs with
property (SR).
For a graph G, by P (G; x) we denote the characteristic polynomial of A(G). If S
is a set of vertices and edges in G,by G − S we mean the graph obtained by deleting
all the elements of S from G. It is understood that when a vertex is deleted,all edges
incident with it are deleted as well,but when an edge is deleted,the vertices incident
with it are not. For a vertex v in G, d(v) denotes the degree of v in G.
If G1 = (V1, E1) and G2 = (V2, E2) be two graphs on disjoint sets of m and n
vertices,respectively,their union is the graph G1 ∪ G2 = (V1 ∪ V2, E1 ∪ E2).
Let G be a graph. A linear subgraph L of G is a disjoint union of some edges and
some cycles in G. A k-matching M in G is a disjoint union of k edges. If e1, e2, . . . , ek
are the edges of a k-matching M ,then we write M = {e1, e2, . . . , ek}. If 2k is the
order of G,then a k-matching of G is called a perfect matching of G.
Let G be a graph on n vertices. Let
P (G; x) = a0xn + a1xn−1 + · · · + an,
be the characteristic polynomial of A(G). Then a0(G) = 1, a1(G) = 0 and −a2(G) is
the numb (...truncated)