scholarly journals Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1663
Author(s):  
Alexander Farrugia
Keyword(s):  
Characteristic Polynomial ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Reconstruction Problem ◽  
Vertex Set ◽  
Disconnected Graphs ◽  
Polynomial Reconstruction ◽  
Standing Problem

Let G be a simple graph and {1,2,…,n} be its vertex set. The polynomial reconstruction problem asks the question: given a deck P(G) containing the n characteristic polynomials of the vertex deleted subgraphs G−1, G−2, …, G−n of G, can ϕ(G,x), the characteristic polynomial of G, be reconstructed uniquely? To date, this long-standing problem has only been solved in the affirmative for some specific classes of graphs. We prove that if there exists a vertex v such that more than half of the eigenvalues of G are shared with those of G−v, then this fact is recognizable from P(G), which allows the reconstruction of ϕ(G,x). To accomplish this, we make use of determinants of certain walk matrices of G. Our main result is used, in particular, to prove that the reconstruction of the characteristic polynomial from P(G) is possible for a large subclass of disconnected graphs, strengthening a result by Sciriha and Formosa.

scholarly journals Properties of the characteristic polynomials and spectrum of Pn and Cn

2016 ◽  
Vol 5 (2) ◽  
pp. 132
Author(s):  
Essam El Seidy ◽  
Salah Eldin Hussein ◽  
Atef Mohamed
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Signless Laplacian Matrix ◽  
Path Graph ◽  
Signless Laplacian ◽  
Vertex Set ◽  
Cycle Graph

We consider a finite undirected and connected simple graph  with vertex set  and edge set .We calculated the general formulas of the spectra of a cycle graph and path graph. In this discussion we are interested in the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and seidel adjacency matrix.

Polynomial reconstruction of signed graphs whose least eigenvalue is close to -2

2016 ◽  
Vol 31 ◽  
pp. 740-753 ◽  
Author(s):  
Slobodan Simić ◽  
Zoran Stanic
Keyword(s):  
Characteristic Polynomial ◽  
Reconstruction Problem ◽  
Signed Graphs ◽  
Simple Graphs ◽  
Least Eigenvalue ◽  
Polynomial Reconstruction ◽  
Special Classes

The polynomial reconstruction problem for simple graphs has been considered in the literature for more than forty years and is not yet resolved except for some special classes of graphs. Recently, the same problem has been put forward for signed graphs. Here, the reconstruction of the characteristic polynomial of signed graphs whose vertex-deleted subgraphs have least eigenvalue greater than $-2$ is considered.

scholarly journals ON SOME BOUNDS OF THE MINIMUM EDGE DOMINATING ENERGY OF A GRAPH

2021 ◽  
Vol 12 (4) ◽  
pp. 1394-1399
Author(s):  
A.Sharmila , Et. al.
Keyword(s):  
Characteristic Polynomial ◽  
Dominating Set ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Eigen Values ◽  
Vertex Set ◽  
G 12

Let G be a simple graph of order n with vertex set V= {v1, v2, ..., vn} and edge set  E = {e1, e2, ..., em}. A subset  of E is called an edge dominating set of G if every edge of  E -  is adjacent to some edge in  .Any edge dominating set with minimum cardinality is called a minimum edge dominating set [2]. Let  be a minimum edge dominating set of a graph G. The minimum edge dominating matrix of G is the m x m matrix defined by G)= , where  = The characteristic polynomial of is denoted by fn (G, ρ) = det (ρI -  (G) ).  The minimum edge dominating eigen values of a graph G are the eigen values of (G).  Minimum edge dominating energy of G is defined as                 (G) =   [12] In this paper we have computed the Minimum Edge Dominating Energy of a graph. Its properties and bounds are discussed. All graphs considered here are simple, finite and undirected.

scholarly journals Joining Forces for Reconstruction Inverse Problems

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1687
Author(s):  
Irene Sciriha
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Graph Matching ◽  
Isomorphism Problem ◽  
Reconstruction Problem ◽  
Graph Invariants ◽  
Minimal Set ◽  
Disconnected Graphs ◽  
Polynomial Reconstruction ◽  
Spectral Inverse Problem

A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP.

scholarly journals Tree-Antimagicness of Disconnected Graphs

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Andrea Semaničová-Feňovčíková ◽  
Muhammad Awais Umar
Keyword(s):  
Arithmetic Progression ◽  
Disjoint Union ◽  
Simple Graph ◽  
Initial Term ◽  
Vertex Set ◽  
The Common ◽  
Disconnected Graphs

A simple graphGadmits anH-covering if every edge inE(G)belongs to a subgraph ofGisomorphic toH. The graphGis said to be (a,d)-H-antimagic if there exists a bijection from the vertex setV(G)and the edge setE(G)onto the set of integers1, 2, …,VG+E(G)such that, for all subgraphsH′ofGisomorphic toH, the sum of labels of all vertices and edges belonging toH′constitute the arithmetic progression with the initial termaand the common differenced.Gis said to be a super (a,d)-H-antimagic if the smallest possible labels appear on the vertices. In this paper, we study super tree-antimagic total labelings of disjoint union of graphs.

Arithmetic–geometric energy of specific graphs

2020 ◽  
pp. 2150005
Author(s):  
Lu Zheng ◽  
Gui-Xian Tian ◽  
Shu-Yu Cui
Keyword(s):  
Characteristic Polynomial ◽  
Geometric Characteristic ◽  
Simple Graph ◽  
Energy Change ◽  
Vertex Set ◽  
Energy Formula ◽  
Matrix Formula

Let [Formula: see text] be a simple graph of order [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text]. The arithmetic–geometric matrix [Formula: see text] of [Formula: see text] is a matrix of order [Formula: see text] defined by [Formula: see text] if [Formula: see text] and 0 otherwise, where [Formula: see text] stands for the degree of the vertex [Formula: see text] in [Formula: see text]. The arithmetic–geometric characteristic polynomial of [Formula: see text] is the characteristic polynomial of [Formula: see text]. The arithmetic–geometric energy [Formula: see text] of [Formula: see text] is the sum of absolute values of all eigenvalues of [Formula: see text]. In this paper, we obtain the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy of some specific graphs. In addition, we also consider the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy change of these graphs when one edge is deleted.

scholarly journals The characteristic polynomial of a graph is reconstructible from the characteristic polynomials of its vertex-deleted subgraphs and their complements

10.37236/1490 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Elias M. Hagos
Keyword(s):  
Characteristic Polynomial ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Graph Reconstruction ◽  
The Many

The question of whether the characteristic polynomial of a simple graph is uniquely determined by the characteristic polynomials of its vertex-deleted subgraphs is one of the many unresolved problems in graph reconstruction. In this paper we prove that the characteristic polynomial of a graph is reconstructible from the characteristic polynomials of the vertex-deleted subgraphs of the graph and its complement.

scholarly journals Resistance Distance and Kirchhoff Index of Graphs with Pockets

2018 ◽  
Author(s):  
Qun Liu ◽  
Jiabao Liu
Keyword(s):  
Closed Form ◽  
Simple Graph ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Vertex Set

Let G[F,Vk, Huv] be the graph with k pockets, where F is a simple graph of order n ≥ 1,Vk= {v1,v2,··· ,vk} is a subset of the vertex set of F and Hvis a simple graph of order m ≥ 2,v is a specified vertex of Hv. Also let G[F,Ek, Huv] be the graph with k edge pockets, where F is a simple graph of order n ≥ 2, Ek= {e1,e2,···ek} is a subset of the edge set of F and Huvis a simple graph of order m ≥ 3, uv is a specified edge of Huvsuch that Huv− u is isomorphic to Huv− v. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of G[F,Vk, Hv] and G[F,Ek, Huv] in terms of the resistance distance and Kirchhoff index F, Hv and F, Huv, respectively.

scholarly journals On k-rainbow domination in middle graphs

2021 ◽  
Author(s):  
Kijung Kim
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Matching Number ◽  
Domatic Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Middle Graph ◽  
Dominating Function

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.

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