Unicyclic graphs with strong reciprocal eigenvalue property

Electronic Journal of Linear Algebra, Sep 2017

By Sasmita Barik, Milan Nath, Sukanta Pati, et al., Published on 01/01/08

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Unicyclic graphs with strong reciprocal eigenvalue property

Volume 1081-3810 Unicyclic graphs with strong reciprocal eigenvalue property Sasmita Barik Bhaba Kumar Sarma Follow this and additional works at: https://repository.uwyo.edu/ela Recommended Citation - http://math.technion.ac.il/iic/ela 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)


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Sasmita Barik, Milan Nath, Sukanta Pati, Bhaba Kumar Sarma. Unicyclic graphs with strong reciprocal eigenvalue property, Electronic Journal of Linear Algebra, 2018, Volume 17, Issue 1,