Nonsingular Graph Research Articles

Singular graphs and the reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be its adjacency matrix. The terms singularity, eigenvalues, and characteristic polynomial of G mean those of A(G). A nonsingular graph G is said to have the reciprocal eigenvalue property if the reciprocal of each eigenvalue of G is also an eigenvalue. A graph G (possibly singular) is said to have the weak reciprocal eigenvalue property if the reciprocal of each nonzero eigenvalue of it is also an eigenvalue. In Barik et al. (2022) [3], the authors proved that there is no nontrivial tree with the weak reciprocal eigenvalue property and posed the following question: “Does there exist a nontrivial graph with the weak reciprocal eigenvalue property?” Suppose that G is singular and the characteristic polynomial of G is xn−k(xk+a1xk−1+⋯+ak). Assume that A(G) has rank k, so that ak≠0. Can we ever have |ak|=1? The answer turns out to be negative. As an application, we settle the question posed in Barik et al. (2022) [3]. Another similar application is also mentioned. It is natural to wonder, “Does there exist a nontrivial, simple, connected weighted graph with the weak reciprocal eigenvalue property?” We provide a class of such graphs. Furthermore, we extend our results to weighted graphs.

Conversion of Unweighted Graphs to Weighted Graphs Satisfying Properties R and −SR

Spectral graph theory is like a special tool for understanding graphs. It helps us find patterns and connections in complex networks, using the magic of eigenvalues. Let G be the graph and A(G) be its adjacency matrix, then G is singular if the determinant of the adjacency matrix A(G) is 0, otherwise it is nonsingular. Within the realm of nonsingular graphs, there is the concept of property R, where each eigenvalue’s reciprocal is also an eigenvalue of G. By introducing multiplicity constraints on both eigenvalues and their reciprocals, it becomes property SR. Similarly, the world of nonsingular graphs reveals property −R, where the negative reciprocal of each eigenvalue also finds a place within the spectrum of G. Moreover, when the multiplicity of each eigenvalue and its negative reciprocal is equal, this results in a graph with a property of −SR. Some classes of unweighted nonbipartite graphs are already constructed in the literature with the help of the complete graph Kn and a copy of the path graph P4 satisfying property R but not SR. This article takes this a step further. The main aim is to construct several weighted classes of graphs which satisfy property R but not SR. For this purpose, the weight functions are determined that enable these nonbipartite graph classes to satisfy the −SR and R properties, even if the unweighted graph does not satisfy these properties. Some examples are presented to support the investigated results. These examples explain how certain weight functions make these special types of graphs meet the properties R or −SR, even when the original graphs without weights do not meet these properties.

Quadratic embedding constants of graphs: Bounds and distance spectra

The quadratic embedding constant (QEC) of a finite, simple, connected graph G is the maximum of the quadratic form of the distance matrix of G on the subset of the unit sphere orthogonal to the all-ones vector. The study of these QECs was motivated by the classical work of Schoenberg on quadratic embedding of metric spaces [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938]. In this article, we provide sharp upper and lower bounds for the QEC of trees. Next, we identify a new subclass of nonsingular graphs whose QEC is the second largest distance eigenvalue. Finally, we show that the QEC of the cluster of an arbitrary graph G with either a complete or star graph can be computed in terms of the QEC of G. As an application of this result, we provide new families of examples of graphs of QE class.

Spectral Characterization of Graphs with Respect to the Anti-Reciprocal Eigenvalue Property

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, −1λ∈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 (−SR) graph). In this article, a new family of graphs Fn(k,j) is introduced and the energy of F5(k,k2)k≥2 is calculated. Furthermore, with the help of F5(k,k2), some families of (−SR) graphs are constructed.

On weighted noncorona graphs with property R and −SR

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.

Energy of Nonsingular Graphs: Improving Lower Bounds

Let G be a simple graph of order n and A be its adjacency matrix. Let λ 1 ≥ λ 2 ≥ … ≥ λ n be eigenvalues of matrix A . Then, the energy of a graph G is defined as ε G = ∑ i = 1 n λ i . In this paper, we will discuss the new lower bounds for the energy of nonsingular graphs in terms of degree sequence, 2-sequence, the first Zagreb index, and chromatic number. Moreover, we improve some previous well-known bounds for connected nonsingular graphs.

A Geometric Property of the Laplacian matrix of a Connected Nonsingular Mixed Graph

In this paper, we give a geometric interpretation of the Laplacian matrix of a connected nonsingular mixed graph which generalizes the results of M. Fiedler (M. Fiedler, Geometry of the Laplacian, Linear Algebra Appl., 2005, 403: 409–413). In addition, the relations of geometric properties between a connected (singular or nonsingular) mixed graph, and all its resigned graphs will be characterized.

Construction of larger singular and nonsingular graphs using a path

Construction of larger singular and nonsingular graphs using a path

On the isomorphisms between evolution algebras of graphs and random walks

Evolution algebras are non-associative algebras inspired from biological phenomena, with applications to or connections with different mathematical fields. There are two natural ways to define an evolution algebra associated to a given graph. While one takes into account only the adjacencies of the graph, the other includes probabilities related to the symmetric random walk on the same graph. In this work we state new properties related to the relation between these algebras, which is one of the open problems in the interplay between evolution algebras and graphs. On the one hand, we show that for any graph both algebras are strongly isotopic. On the other hand, we provide conditions under which these algebras are or are not isomorphic. For the case of finite non-singular graphs we provide a complete description of the problem, while for the case of finite singular graphs we state a conjecture supported by examples and partial results. The case of graphs with an infinite number of vertices is also discussed. As a sideline of our work, we revisit a result existing in the literature about the identification of the automorphism group of an evolution algebra, and we give an improved version of it.

Two-graphs and NSSDs: An algebraic approach

A two-graph (V,Δ) is a combinatorial entity consisting of a set V together with a collection Δ of unordered triples of elements of V, such that there exists a graph G with vertex set V in which the triples in Δ are precisely the subgraphs K3 or K2∪̇K1 induced in G. Switching is an equivalence relation partitioning the graphs on n vertices into switching classes. All the graphs that are switching equivalent to G yield the same Δ and therefore the two–graph (V,Δ) is taken to consist of the graphs in the switching class of G. The graphs in a switching class have similar (0,±1)-Seidel matrices. So the spectrum of (V,Δ) is taken as the Seidel spectrum of a graph in the switching class. The two graphs with exactly two distinct Seidel eigenvalues μ1,μ2 are regular. We show how an involution M(μ1,μ2) provides a simple way to determine structural and combinatorial properties of the graphs of a regular two-graph. If μ1+μ2=0, the regular two-graph consists of conference graphs. We show that M(μ1,−μ1) is an NSSD (non-singular graph with a singular deck) with the special property of being a nutful graph. The rich properties of a nutful NSSD reveal new spectral properties of conference graphs.

Edge construction of molecular NSSDs

A nonsingular graph with a singular deck, or NSSD, is a graph with weighted edges and no loops whose adjacency matrix is nonsingular and whose vertex-deleted subgraphs have singular adjacency matrices. When all the weights of the edges of a NSSD are equal to one, we obtain a molecular NSSD, which is an ipso omni-insulator molecular graph. We present necessary and sufficient conditions for a NSSD G to remain a NSSD after increasing the weight of one of its edges by w∈R∖{0}; this weight change may result in the addition or removal of that edge to or from G. Using this result, we construct molecular NSSDs of a fixed even order. This is accomplished by systematically introducing edges to, or removing edges from, a nonsingular tree.

English

A graph is nonsingular if its adjacency matrix A(G) is nonsingular. The inverse of a nonsingular graph G is a graph whose adjacency matrix is similar to A(G)−1 via a particular type of similarity. Let H denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in H which possess unicyclic inverses. We present a characterization of unicyclic graphs in H which possess bicyclic inverses.

Self-inverse unicyclic graphs and strong reciprocal eigenvalue property

We consider only simple graphs. A graph G is said to be nonsingular if its adjacency matrix A(G) is nonsingular. The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of ±1s. An invertible graph G is said to be a self-inverse graph if G is isomorphic to its inverse. Let H be the class of connected bipartite graphs with unique perfect matchings. We consider the problem of characterizing the graphs in H which are self-inverse graphs. In this article, we answer this problem for the class of unicyclic graphs.The study of the strong reciprocal eigenvalue property is closely related to the study of graphs which are isomorphic to their inverse graphs. A nonsingular graph G is said to satisfy the property (SR) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and they both have the same multiplicities. In this article, we show that if a unicyclic graph G in H has property (SR), then G is invertible and the inverse graph of G is unicyclic. As an application, we show that a noncorona unicyclic graph in H with property (SR) can have one of the five specified structures. Finally, our discussions lead to the following problem. Does there exist a unicyclic graph G∈H which has property (SR) but G is not self-inverse?

Triangles in inverse NSSD graphs

A nonsingular graph with a singular deck, or NSSD, is an edge-weighted graph with all the diagonal entries of its adjacency matrix and of the inverse equal to zero. We obtain necessary and sufficient conditions for a two-vertex-deleted subgraph of an NSSD G to remain an NSSD by considering triangles in the inverse NSSD whose adjacency matrix is . We also consider the graph obtained by introducing terminal vertices to the NSSD G and show that the conditions for it to be an NSSD are associated with the conditions deduced for the two-vertex deleted subgraph.

Inverses of bicyclic graphs

A graph G is said to be nonsingular (resp., singular) if its adjacency matrix A(G) is nonsingular (resp., singular). The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of ±1s. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite. In [C.D. Godsil. Inverses of trees. Combinatorica, 5(1):33–39, 1985.], Godsil proved that such graphs are invertible. He posed the question of characterizing the bipartite graphs with unique perfect matchings possessing inverses. In this article, Godsil’s question for the class of bicyclic graphs is answered.

Some new lower bounds for energy of graphs

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. In this paper we present some new lower bounds for energy of non-singular graphs, connected non-singular graphs and connected unicyclic non-singular graphs in terms of number of vertices, number of edges, maximum degree and Zagreb indices.

Graphs with reciprocal eigenvalue properties

In this paper, only simple graphs are considered. A graph G is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G satisfies reciprocal eigenvalue property (property R) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property (property SR) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and they both have the same multiplicities. From the definitions property SR implies property R. Furthermore, for some classes of graphs (for example, trees), it is known that these properties are equivalent. However, the equivalence of these two properties is not yet known for any nonsingular graph. In this article, it is shown that these properties are not equivalent in general.

On some graphs which possess inverses

A graph is nonsingular if its adjacency matrix is nonsingular. A nonsingular graph is said to have an inverse if is signature similar to a nonnegative matrix. An -graph is a bipartite graph with a unique perfect matching . We introduce the ‘even’ness of a nonmatching edge and show that under certain conditions a connected -graph has an inverse if and only if is bipartite, where is the set of all even nonmatching edges. This extends some known results, providing us with a larger class of graphs possessing inverses.

On The Inverse Of A Class Of Bipartite Graphs With Unique Perfect Matchings

Let G be a simple, undirected graph and Gw be the weighted graph obtained from G by giving weights to its edges using a positive weight function w. A weighted graph Gw is said to be nonsingular if its adjacency matrix A(Gw) is nonsingular. In [9], Godsil has given a class $\mathcal{G }$of connected, unweighted, bipartite, nonsingular graphs G with a unique perfect matching, such that A(G)−1 is signature similar to a nonnegative matrix, that is, there exists a diagonal matrix D with diagonal entries ±1 such that DA(G)−1D is nonnegative. The graph associated to the matrix DA(G)−1D is called the inverse of G and it is denoted by G+. The graph G+ is an undirected, weighted, connected, bipartite graph with a unique perfect matching. Nonsingular, unweighted trees are contained inside the class G. We first give a constructive characterization of the class of weighted graphs Hw that can occur as the inverse of some graph G∈\mathcal{ G}. This generalizes Theorem 2.6 of Neumann and Pati[13], where the authors have characterized graphs that occur as inverses of nonsingular, unweighted trees. A weighted graph Gw is said to have the property (R) if for each eigenvalue λ of A(Gw), 1⁄λ is also an eigenvalue of A(Gw). If further, the multiplicity of λ and 1⁄λ are the same, then Gw is said to have property (SR). A characterization of the class of nonsingular, weighted trees Tw with at least 8 vertices that have property (R) was given in [13] under some restriction on the weights. It is natural to ask for such a characterization for the whole of G, possibly with some weaker restrictions on the weights. We supply such a characterization. In particular, for trees it settles an open problem raised in [13].

Non-Singular graphs with a Singular Deck

The n-vertex graph G(=Γ(G)) with a non-singular real symmetric adjacency matrix G, having a zero diagonal and singular (n−1)×(n−1) principal submatrices is termed a NSSD, a Non-Singular graph with a Singular Deck. NSSDs arose in the study of the polynomial reconstruction problem and were later found to characterise non-singular molecular graphs that are distinct omni-conductors and ipso omni-insulators. Since both matrices G and G−1 represent NSSDs Γ(G) and Γ(G−1), the value of the nullity of a one-, two- and three-vertex deleted subgraph of G is shown to be determined by the corresponding subgraph in Γ(G−1). Constructions of infinite subfamilies of non-NSSDs are presented. NSSDs with all two-vertex deleted subgraphs having a common value of the nullity are referred to as G-nutful graphs. We show that their minimum vertex degree is at least 4.