A new graph radio k-coloring algorithm

2019 ◽  
Vol 11 (01) ◽  
pp. 1950005 ◽  
Author(s):  
Laxman Saha ◽  
Pratima Panigrahi
Keyword(s):  
Connected Graph ◽  
Channel Assignment ◽  
Time Complexity ◽  
Radio Spectrum ◽  
Circulant Graph ◽  
Communication Services ◽  
Channel Assignment Problem ◽  
Vertex Set ◽  
General Graphs ◽  
Simple Connected Graph

Due to the rapid growth in the use of wireless communication services and the corresponding scarcity and the high cost of radio spectrum bandwidth, Channel assignment problem (CAP) is becoming highly important. Radio [Formula: see text]-coloring of graphs is a variation of CAP. For a positive integer [Formula: see text], a radio [Formula: see text]-coloring of a simple connected graph [Formula: see text] is a mapping [Formula: see text] from the vertex set [Formula: see text] to the set [Formula: see text] of non-negative integers such that [Formula: see text] for each pair of distinct vertices [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the distance between [Formula: see text] and [Formula: see text] in [Formula: see text]. The span of a radio [Formula: see text]-coloring [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text] and the radio[Formula: see text]-chromatic number of [Formula: see text], denoted by [Formula: see text], is [Formula: see text] where the minimum is taken over all radio [Formula: see text]-coloring of [Formula: see text]. In this paper, we present two radio [Formula: see text]-coloring algorithms for general graphs which will produce radio [Formula: see text]-colorings with their spans. For an [Formula: see text]-vertex simple connected graph the time complexity of the both algorithm is of [Formula: see text]. Implementing these algorithms we get the exact value of [Formula: see text] for several graphs (for example, [Formula: see text], [Formula: see text], [Formula: see text], some circulant graph etc.) and many values of [Formula: see text], especially for [Formula: see text].

General eccentric distance sum of graphs

2020 ◽  
pp. 2150046
Author(s):  
Tomáš Vetrík
Keyword(s):  
Connected Graph ◽  
Bipartite Graphs ◽  
Extremal Graphs ◽  
Vertex Connectivity ◽  
Vertex Set ◽  
General Graphs ◽  
Eccentric Distance

For [Formula: see text], we define the general eccentric distance sum of a connected graph [Formula: see text] as [Formula: see text], where [Formula: see text] is the vertex set of [Formula: see text], [Formula: see text] is the eccentricity of a vertex [Formula: see text] in [Formula: see text], [Formula: see text] and [Formula: see text] is the distance between vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. This index generalizes several other indices of graphs. We present some bounds on the general eccentric distance sum for general graphs, bipartite graphs and trees of given order, graphs of given order and vertex connectivity and graphs of given order and number of pendant vertices. The extremal graphs are presented as well.

scholarly journals Zagreb, multiplicative zagreb indices and coindices of 〖nc〗_n (k) and 〖ca〗_3 (c_6 ) graphs

2018 ◽  
Vol 36 (2) ◽  
pp. 9-15
Author(s):  
Vida Ahmadi ◽  
Mohammad Reza Darafshe
Keyword(s):  
Connected Graph ◽  
Zagreb Indices ◽  
The Third ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Simple Connected Graph

Let  be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are defind, respectivly by: ,   and   where  is the degree of vertex u in G and uv is an edge of G, connecting the vertices u and v. Recently, the first and second multiplicative Zagreb indices of graph  are defind by:  and . The first and second Zagreb coindices of graph are defind by:  and .  and , named as multiplicative Zagreb coindices. In this article, we compute the first, second and the third Zagreb indices and the first and second multiplicative Zagreb indices of some  graphs. The first and second Zagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.

scholarly journals Rectilinear Monotone r-Regular Planar Graphs for r = {3, 4, 5}

d'CARTESIAN ◽  
2015 ◽  
Vol 4 (1) ◽  
pp. 103
Author(s):  
Arthur Wulur ◽  
Benny Pinontoan ◽  
Mans Mananohas
Keyword(s):  
Planar Graph ◽  
Connected Graph ◽  
Planar Graphs ◽  
Infinite Family ◽  
Regular Graphs ◽  
Vertex Set ◽  
Simple Connected Graph

A graph G consists of non-empty set of vertex/vertices (also called node/nodes) and the set of lines connecting two vertices called edge/edges. The vertex set of a graph G is denoted by V(G) and the edge set is denoted by E(G). A Rectilinear Monotone r-Regular Planar Graph is a simple connected graph that consists of vertices with same degree and horizontal or diagonal straight edges without vertical edges and edges crossing. This research shows that there are infinite family of rectilinear monotone r-regular planar graphs for r = 3and r = 4. For r = 5, there are two drawings of rectilinear monotone r-regular planar graphs with 12 vertices and 16 vertices. Keywords: Monotone Drawings, Planar Graphs, Rectilinear Graphs, Regular Graphs

scholarly journals Edge metric dimension of some classes of circulant graphs

2020 ◽  
Vol 28 (3) ◽  
pp. 15-37
Author(s):  
Muhammad Ahsan ◽  
Zohaib Zahid ◽  
Sohail Zafar
Keyword(s):  
Connected Graph ◽  
Ordered Set ◽  
Metric Dimension ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Metric Basis

AbstractLet G = (V (G), E(G)) be a connected graph and x, y ∈ V (G), d(x, y) = min{ length of x − y path } and for e ∈ E(G), d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e1 and e2, if d(e1, x) ≠ d(e2, x). Let WE = {w1, w2, . . ., wk} be an ordered set in V (G) and let e ∈ E(G). The representation r(e | WE) of e with respect to WE is the k-tuple (d(e, w1), d(e, w2), . . ., d(e, wk)). If distinct edges of G have distinct representation with respect to WE, then WE is called an edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). The circulant graph Cn(1, m) has vertex set {v1, v2, . . ., vn} and edge set {vivi+1 : 1 ≤ i ≤ n−1}∪{vnv1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs Cn(1, 2) and Cn(1, 3) is constant.

scholarly journals New bounds on the rainbow domination subdivision number

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 615-622 ◽  
Author(s):  
Mohyedin Falahat ◽  
Seyed Sheikholeslami ◽  
Lutz Volkmann
Keyword(s):  
Connected Graph ◽  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Simple Connected Graph ◽  
Domination Subdivision Number ◽  
Dominating Function

A 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2} such that for any vertex v ? V(G) with f (v) = ? the condition Uu?N(v) f(u)= {1,2} is fulfilled, where N(v) is the open neighborhood of v. The weight of a 2RDF f is the value ?(f) = ?v?V |f(v)|. The 2-rainbow domination number of a graph G, denoted by r2(G), is the minimum weight of a 2RDF of G. The 2-rainbow domination subdivision number sd?r2(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the 2-rainbow domination number. In this paper we prove that for every simple connected graph G of order n ? 3, sd?r2(G)? 3 + min{d2(v)|v?V and d(v)?2} where d2(v) is the number of vertices of G at distance 2 from v.

scholarly journals The Differential on Graph Operator Q(G)

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 751
Author(s):  
Ludwin Basilio ◽  
Jair Simon ◽  
Jesús Leaños ◽  
Omar Cayetano
Keyword(s):  
Connected Graph ◽  
Independence Number ◽  
Vertex Cover ◽  
Domination Number ◽  
Graph Invariants ◽  
Maximum Value ◽  
Vertex Set ◽  
Simple Connected Graph

If G = ( V ( G ) , E ( G ) ) is a simple connected graph with the vertex set V ( G ) and the edge set E ( G ) , S is a subset of V ( G ) , and let B ( S ) be the set of neighbors of S in V ( G ) ∖ S . Then, the differential of S ∂ ( S ) is defined as | B ( S ) | − | S | . The differential of G, denoted by ∂ ( G ) , is the maximum value of ∂ ( S ) for all subsets S ⊆ V ( G ) . The graph operator Q ( G ) is defined as the graph that results by subdividing every edge of G once and joining pairs of these new vertices iff their corresponding edges are incident in G. In this paper, we study the relations between ∂ ( G ) and ∂ ( Q ( G ) ) . Besides, we exhibit some results relating the differential ∂ ( G ) and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.

THE SIGNED STAR DOMINATION NUMBER OF CAYLEY GRAPHS

2012 ◽  
Vol 04 (02) ◽  
pp. 1250017 ◽  
Author(s):  
T. TAMIZH CHELVAM ◽  
G. KALAIMURUGAN ◽  
WELL Y. CHOU
Keyword(s):  
Connected Graph ◽  
Cayley Graphs ◽  
Domination Number ◽  
Vertex Set ◽  
Simple Connected Graph ◽  
Dominating Function

Let G be a simple connected graph with vertex set V(G) and edge set E(G). A function f : E(G) → {-1, 1} is called a signed star dominating function (SSDF) on G if ∑e∈E(v) f(e) ≥ 1 for every v ∈ V(G), where E(v) is the set of all edges incident to v. The signed star domination number of G is defined as γ SS (G) = min {∑e∈E(G) f(e) | f is a SSDF on G}. In this paper, we obtain exact values for the signed star domination number for certain classes of Cayley digraphs and Cayley graphs.

scholarly journals Π(G,x) Polynomial and Π(G) Index of Armchair Polyhex Nanotubes TUAC6[m,n]

2014 ◽  
Vol 36 ◽  
pp. 201-206
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Connected Graph ◽  
Infinite Class ◽  
Vertex Set ◽  
Omega Polynomial ◽  
Multiple Edges ◽  
Simple Connected Graph

Let G be a simple connected graph with the vertex set V = V(G) and the edge set E = E(G), without loops and multiple edges. For counting qoc strips in G, Omega polynomial was introduced by Diudea and was defined as Ω(G,x) = ∑cm(G,c)xc where m(G,c) be number of qoc strips of length c in the graph G. Following Omega polynomial, the Sadhana polynomial was defined by Ashrafi et al as Sd(G,x) = ∑cm(G,c)x[E(G)]-c in this paper we compute the Pi polynomial Π(G,x) = ∑cm(G,c)x[E(G)]-c and Pi Index Π(G ) = ∑cc·m(G,c)([E(G)]-c) of an infinite class of “Armchair polyhex nanotubes TUAC6[m,n]”.

scholarly journals On the average of the eccentricities of a graph

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1395-1401 ◽  
Author(s):  
Kinkar Das ◽  
Kexiang Xu ◽  
Xia Li ◽  
Haiqiong Liu
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Mean Value ◽  
Extremal Graphs ◽  
The Mean ◽  
Average Eccentricity ◽  
General Graphs ◽  
Simple Connected Graph

Let G = (V; E) be a simple connected graph of order n with m edges. Also let eG(vi) be the eccentricity of a vertex vi in G. We can assume that eG(v1) eG(v2) ? ... ? eG(vn-1) ? eG(vn). The average eccentricity of a graph G is the mean value of eccentricities of vertices of G, avec(G) = 1/n ?n,i=1 eG(vi). Let ? = ?G be the largest positive integer such that eG(vG ) ? avec(G). In this paper, we study the value of G of a graph G. For any tree T of order n, we prove that 2 ? ?T ? n - 1 and we characterize the extremal graphs. Moreover, we prove that for any graph G of order n,2 ? ?G ? n and we characterize the extremal graphs. Finally some Nordhaus-Gaddum type results are obtained on ?G of general graphs G.

The eccentric adjacency index of unicyclic graphs and trees

2018 ◽  
Vol 13 (01) ◽  
pp. 2050028 ◽  
Author(s):  
Shehnaz Akhter ◽  
Rashid Farooq
Keyword(s):  
Connected Graph ◽  
Connectivity Index ◽  
Eccentric Connectivity Index ◽  
Unicyclic Graphs ◽  
Vertex Set ◽  
Simple Connected Graph ◽  
Eccentric Adjacency Index

Let [Formula: see text] be a simple connected graph with vertex set [Formula: see text] and edge set [Formula: see text]. The eccentricity [Formula: see text] of a vertex [Formula: see text] in [Formula: see text] is the largest distance between [Formula: see text] and any other vertex of [Formula: see text]. The eccentric adjacency index (also known as Ediz eccentric connectivity index) of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the sum of degrees of neighbors of the vertex [Formula: see text]. In this paper, we determine the unicyclic graphs with largest eccentric adjacency index among all [Formula: see text]-vertex unicyclic graphs with a given girth. In addition, we find the tree with largest eccentric adjacency index among all the [Formula: see text]-vertex trees with a fixed diameter.

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