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Let A(G) be the adjacency matrix of a simple connected undirected graph G. A graph G of order n is said to be non-singular (respectively singular) if A(G) is non-singular (respectively singular). The spectrum of a graph G is the set of all its eigenvalues denoted by spec(G). The antireciprocal (respectively reciprocal) eigenvalue property for a graph G can be defined as "Let G be a non-singular graph G if the negative reciprocal (respectively positive reciprocal) of each eigenvalue is likewise an eigenvalue of G, then G has anti-reciprocal (respectively reciprocal) eigenvalue property." Furthermore, a graph G is said to have strong anti-reciprocal eigenvalue property (resp. strong reciprocal eigenvalue property) if the eigenvalues and their negative (resp. positive) reciprocals are of same multiplicities. In this article, graphs satisfying anti-reciprocal eigenvalue (or property (-R)) and strong anti-reciprocal eigenvalue property (or property (-SR)) are discussed. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple connected graph with vertex set V and edge set E, respectively. The term "anti-reciprocal eigenvalue property" refers to a non-singular graph G for which, − 1 λ ∈ σ (G) , whenever λ ∈ σ (G) , ∀ λ ∈ σ (G) . Here, σ (G) is the multiset of all eigenvalues of A (G) . Moreover, if multiplicities of eigenvalues and their negative reciprocals are equal, then that graph is said to have strong anti-reciprocal eigenvalue properties, and the graph is referred to as a strong anti-reciprocal graph (or (− S R) graph). In this article, a new family of graphs F n (k , j) is introduced and the energy of F 5 (k , k 2) k ≥ 2 is calculated. Furthermore, with the help of F 5 (k , k 2) , some families of (− S R) graphs are constructed. [ABSTRACT FROM AUTHOR]

Let Gw be a simple weighted graph with adjacency matrix A(Gw). The set of all eigenvalues of A(Gw) is called the spectrum of weighted graph Gw denoted by σ(Gw). The reciprocal eigenvalue property (or property R) for a connected weighted nonsingular graph Gw is defined as, if η ∈ σ(Gw) then 1 η ∈ σ(Gw). Further, if η and 1 η have the same multiplicities for each η ∈ σ(Gw) then this graph is said to have strong reciprocal eigenvalue property (or property SR). Similarly, a connected weighted nonsingular graph Gw is said to have anti-reciprocal eigenvalue property (or property -R) if η ∈ σ(Gw) then - 1 η ∈ σ(Gw). Furthermore, if η and - 1 η have the same multiplicities for each η ∈ σ(Gw) then strong anti-reciprocal eigenvalue property (or property -SR) holds for the weighted graph Gw. In this article, classes of weighted noncorona graphs satisfying property R and property -SR are studied. [ABSTRACT FROM AUTHOR]