Abstract
In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. Let G 1 , n c and G 2 , n c be the classes of the connected graphs of order n whose complements are bicyclic with exactly two and three cycles, respectively. In this paper, we characterize the unique minimizing graph among all the graphs which belong to G n c = G 1 , n c ∪ G 2 , n c , a class of the connected graphs of order n whose complements are bicyclic.
Highlights
IntroductionIn a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum
Let G be a finite, simple and undirected graph with the vertex-set V ( G ) = {vi : 1 ≤ i ≤ n} and the edge-set E( G ) such that |V ( G )| = n and | E( G )| = m are order and size of the graph G, respectively.The adjacency matrix A( G ) = [ ai,j ] of the graph G is a matrix of order n, where ai,j = 1 if vi is adjacent to v j and ai,j = 0, otherwise
If λ1 ( G ) is the least, one can arrange the eigenvalues as λ1 ( G ) ≤ λ2 ( G ) ≤ ... ≤ λn ( G ), and the eigenvector corresponding to the least eigenvalue is called the first eigenvector
Summary
In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. Mathematics 2017, 5, 18 discussion of the least eigenvalue of the graphs whose complements are connected and contain cliques of small sizes. Fan et al [8] characterized the unique minimizing graph in the class of graphs of order n whose complements are trees. We continue this study and characterize the unique minimizing graph among all the graphs which belong to a class of the connected graphs of order n whose complements are bicyclic with two or three cycles. The rest of the paper is organized as follows: in Section 2, we present some basic definitions and terminologies that are frequently used in the main results and Section 3 includes the main results from the minimizing graph of the connected graphs whose complements are bicyclic
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