On the Stability of Equi Neighbor Polynomial of Graphs
Dhanya P
Department of Mathematics, CKGM Govt. College, Perambra P. O, Kozhikode, Kerala, 673 525, India.
Anil Kumar V
Department of Mathematics, University of Calicut, Malappuram, Kerala, 673 635, India.
Rajeesh C
Department of Mathematics, CKGM Govt. College, Perambra P. O, Kozhikode, Kerala, 673 525, India.
Susanth P
Department of Mathematics, Pookoya Thangal Memorial Government College, Perinthalmanna, Kerala, 679322, India.
Premodkumar K P *
Department of Mathematics, Govt. College Malappuram, Kerala, 676509, India.
*Author to whom correspondence should be addressed.
Abstract
Let Γ(V,E) be a simple graph of order n with vertex set V and edge set E. For an unordered pair of distinct vertices of Γ, we denote (\(x, y\)). For a vertex \(x\) ∈ Γ, let N(\(x\)) be the set of all vertices of Γ that are adjacent to \(x\). Then, for 0 ≤ j ≤ n − 1, the j-equi neighbour set of Γ is defined as Ne(Γ, j) = {(\(x, y\)) \(x, y\) ∈ V, \(x\) ̸= \(y\) and |N (\(x\))| = |N (\(y\))| = j}. The equi-neighbour polynomial of Γ, denoted by Ne[Γ; \(y\)], is given by Ne[Γ; \(y\)] = \(\sum_{j=0}^{n-1}\) |Ne(Γ, j)|\(y\)j . A root of the polynomial Ne[Γ; \(y\)] is defined as the equi neighbour root of the graph Γ. In this work, we study the distribution of zeros of the equi-neighbour polynomial associated with graphs and examine the stability of the common neighbour polynomial relative to the closed right half-plane. We further characterize the conditions under which the common neighbour polynomial for specific classes of graphs satisfies the Hurwitz stability criterion.
Keywords: j− equi neighbour set, equi neighbour polynomial