scholarly journals General Bounds for Identifying Codes in Some Infinite Regular Graphs

10.37236/1583 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Irène Charon ◽  
Iiro Honkala ◽  
Olivier Hudry ◽  
Antoine Lobstein
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Lower And Upper Bounds ◽  
Identifying Codes ◽  
Identifying Code

Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.

scholarly journals Identifying Codes with Small Radius in Some Infinite Regular Graphs

10.37236/1628 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Irène Charon ◽  
Olivier Hudry ◽  
Antoine Lobstein
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Small Radius ◽  
Regular Graphs ◽  
Identifying Codes ◽  
Identifying Code

Let $G=(V,E)$ be a connected undirected graph and $S$ a subset of vertices. If for all vertices $v \in V$, the sets $B_r(v) \cap S$ are all nonempty and different, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $S$ an $r$-identifying code. We give constructive upper bounds on the best possible density of $r$-identifying codes in four infinite regular graphs, for small values of $r$.

scholarly journals Degree Constrained Orientations in Countable Graphs

10.37236/846 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Attila Bernáth ◽  
Henning Bruhn
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lower And Upper Bounds ◽  
Finite Graphs ◽  
Degree Function ◽  
Necessary And Sufficient

Degree constrained orientations are orientations of an (undirected) graph where the in-degree function satisfies given lower and upper bounds. For finite graphs Frank and Gyárfás (1976) gave a necessary and sufficient condition for the existence of such an orientation. We extend their result to countable graphs.

scholarly journals Fault-Tolerant Resolvability and Extremal Structures of Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 78 ◽  
Author(s):  
Hassan Raza ◽  
Sakander Hayat ◽  
Muhammad Imran ◽  
Xiang-Feng Pan
Keyword(s):  
Fault Tolerant ◽  
Upper Bounds ◽  
Metric Dimension ◽  
Regular Graphs ◽  
Lower And Upper Bounds ◽  
Constant Fault

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n − 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure.

scholarly journals On the Minimum Size of an Identifying Code Over All Orientations of a Graph

10.37236/7117 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Nathann Cohen ◽  
Frédéric Havet
Keyword(s):  
Polynomial Time ◽  
Upper Bounds ◽  
Minimum Size ◽  
Np Hard ◽  
Lower And Upper Bounds ◽  
Fixed Integer ◽  
Identifying Code

If $G$ be a graph or a digraph, let $\mathrm{id}(G)$ be the minimum size of an identifying code of $G$ if one exists, and $\mathrm{id}(G)=+\infty$ otherwise. For a graph $G$, let $\mathrm{idor}(G)$ be the minimum of $\mathrm{id}(D)$ overall orientations $D$ of $G$. We give some lower and upper bounds on $\mathrm{idor}(G)$. In particular, we show that $\mathrm{idor}(G)\leqslant \frac{3}{2} \mathrm{id}(G)$ for every graph $G$. We also show that computing $\mathrm{idor}(G)$ is NP-hard, while deciding whether $\mathrm{idor}(G)\leqslant |V(G)|-k$ is polynomial-time solvable for every fixed integer $k$.

scholarly journals On the Randić Index of Corona Product Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
Ismael G. Yero ◽  
Juan A. Rodríguez-Velázquez
Keyword(s):  
Degree Sequence ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Regular Graphs ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Randic Index ◽  
Product Graphs ◽  
Vertex Set ◽  
Corona Product

Let G be a graph with vertex set V=(v1,v2,…,vn). Let δ(vi) be the degree of the vertex vi∈V. If the vertices vi1,vi2,…,vih+1 form a path of length h≥1 in the graph G, then the hth order Randić index Rh of G is defined as the sum of the terms 1/δ(vi1)δ(vi2)⋯δ(vih+1) over all paths of length h contained (as subgraphs) in G. Lower and upper bounds for Rh, in terms of the vertex degree sequence of its factors, are obtained for corona product graphs. Moreover, closed formulas are obtained when the factors are regular graphs.

scholarly journals On the Aα-Eigenvalues of Signed Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1990
Author(s):  
Germain Pastén ◽  
Oscar Rojo ◽  
Luis Medina
Keyword(s):  
Adjacency Matrix ◽  
Undirected Graph ◽  
Upper Bounds ◽  
Signed Graph ◽  
Lower And Upper Bounds ◽  
Signed Graphs ◽  
Basic Properties ◽  
Underlying Graph ◽  
Positive Semidefiniteness ◽  
Vertex Degrees

For α∈[0,1], let Aα(Gσ)=αD(G)+(1−α)A(Gσ), where G is a simple undirected graph, D(G) is the diagonal matrix of its vertex degrees and A(Gσ) is the adjacency matrix of the signed graph Gσ whose underlying graph is G. In this paper, basic properties of Aα(Gσ) are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived—in particular, lower and upper bounds on its largest eigenvalue are obtained.

r-Identifying codes in binary Hamming space, q-ary Lee space and incomplete hypercube

2019 ◽  
Vol 11 (02) ◽  
pp. 1950027
Author(s):  
R. Dhanalakshmi ◽  
C. Durairajan
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Identifying Codes ◽  
Identifying Code ◽  
Construction Techniques ◽  
Lee Metric ◽  
Hamming Space

We study about monotonicity of [Formula: see text]-identifying codes in binary Hamming space, q-ary Lee space and incomplete hypercube. Also, we give the lower bounds for [Formula: see text] where [Formula: see text] is the smallest cardinality among all [Formula: see text]-identifying codes in [Formula: see text] with respect to the Lee metric. We prove the existence of [Formula: see text]-identifying code in an incomplete hypercube. Also, we give the construction techniques for [Formula: see text]-identifying codes in the incomplete hypercubes in Secs. 4.1 and 4.2. Using these techniques, we give the tables (see Tables 1–6) of upper bounds for [Formula: see text] where [Formula: see text] is the smallest cardinality among all [Formula: see text]-identifying codes in an incomplete hypercube with [Formula: see text] processors. Also, we give the exact values of [Formula: see text] for small values of [Formula: see text] and [Formula: see text] (see Sec. 4.3).

scholarly journals Bounds on the $A_{\alpha}$-spread of a graph

2020 ◽  
Vol 36 (36) ◽  
pp. 214-227 ◽  
Author(s):  
Zhen Lin ◽  
Lianying Miao ◽  
Shu-Guang Guo
Keyword(s):  
Real Number ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Upper Bounds ◽  
Diagonal Matrix ◽  
Lower And Upper Bounds ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
The Difference ◽  
The Largest Eigenvalue

Let $G$ be a simple undirected graph. For any real number $\alpha \in[0,1]$, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The $A_{\alpha}$-spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated $A_{\alpha}$-matrix. In this paper, some lower and upper bounds on $A_{\alpha}$-spread are obtained, which extend the results of $A$-spread and $Q$-spread. Moreover, the trees with the minimum and the maximum $A_{\alpha}$-spread are determined, respectively.

scholarly journals Locating-Dominating Sets and Identifying Codes in Graphs of Girth at least 5

10.37236/4562 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Camino Balbuena ◽  
Florent Foucaud ◽  
Adriana Hansberg
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Upper Bounds ◽  
Disjoint Paths ◽  
Minimum Size ◽  
Dominating Sets ◽  
Identifying Codes ◽  
Identifying Code ◽  
Codes In Graphs ◽  
Vertex Disjoint

Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meeting these bounds.

scholarly journals Bounds for Identifying Codes in Terms of Degree Parameters

10.37236/2036 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Florent Foucaud ◽  
Guillem Perarnau
Keyword(s):  
Large Class ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Clique Number ◽  
Minimum Size ◽  
Regular Graphs ◽  
Identifying Codes ◽  
Identifying Code ◽  
Sharp Estimates

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\gamma^{\text{ID}}(G)$ denotes the minimum size of an identifying code of a graph $G$, it was conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there exists a constant $c$ such that if a connected graph $G$ with $n$ vertices and maximum degree $d$ admits an identifying code, then $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{d}+c$. We use probabilistic tools to show that for any $d\geq 3$, $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{\Theta(d)}$ holds for a large class of graphs containing, among others, all regular graphs and all graphs of bounded clique number. This settles the conjecture (up to constants) for these classes of graphs. In the general case, we prove $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{\Theta(d^{3})}$. In a second part, we prove that in any graph $G$ of minimum degree $\delta$ and girth at least 5, $\gamma^{\text{ID}}(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n$. Using the former result, we give sharp estimates for the size of the minimum identifying code of random $d$-regular graphs, which is about $\tfrac{\log d}{d}n$.

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