On Certain Graph Domination Numbers and ... - Semantic Scholar

On Certain Graph Domination Numbers and Applications V. Yegnanarayanan 1 , Valentina E. Balas2 and G. Chitra3 1 2

Professor and HOD, Sciences and Humanities, Vignan University, Guntur-522213, India. Department of Automation, Industrial Engineering, Textiles and Transport, University ”Aurel Vlaicu” Arad, Romania. 3

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Research Scholar, Anna University, Chennai-600025, India. [email protected], 2 [email protected], 3 [email protected]

Abstract: In this paper we compute for paths and cycles certain graph domination invariants like locating domination number, differentiating domination number, global alliance number etc., We also do some comparison analysis of certain parameters defined by combining the domination measures and the second smallest eigen value of the Laplacian matrix of all connected graphs of order 4.While discussing applications we have pointed out the crucial role played by graphs through its hard core structural properties in wireless sensor networks (WSN).

Keywords: Graph, connectivity, domination number, Wireless sensor networks 2010 AMS Subject Classification 05C69, 05C90 Reference to this paper should be made as follows: Yegnanarayanan, V., Valentina Emilia Balas and Chitra.G.(2013) On Certain Graph Domination Numbers and Applications, Int. J. Advanced Intelligence Paradigms, Vol., No., pp.. Biographical notes: Dr. V. Yegnanarayanan finished his Doctoral degree programme in a record period of two and half years and obtained his Ph.D degree in Mathematics from Annamalai University in Oct,1996 by specializing in the area of Graph Theory, a branch of Discrete Mathematics. He has over 25 years of experience in Teaching and Research. He has also worked as Visiting Scientist on lien in esteemed research institutions like Tata Institute of Fundamental Research, Mumbai and Institute of Mathematical Sciences, Chennai. He is a Life Member of ISTE, IMS, RMS and a Senior Member IEEE. He is also serving as a Member of AMS since 2010. He has authored more than 130 Research Papers in referred International/National Journals and Conferences.He Erdos number is 3 and Google Scholar citations as on November 2013 is 152 with H-index 1

No. 7 and i-10 index No.2. Valentina E. Balas is currently Professor in the Department of Automatics and Applied Software at the Faculty of Engineering, University ”Aurel Vlaicu” Arad (Romania). She holds a Ph.D. in Applied Electronics and Telecommunications from Polytechnic University of Timisoara. She is author of more than 160 research papers in refereed journals and International Conferences. Her research interests are in Intelligent Systems, Fuzzy Control, Soft Computing, Smart Sensors, Information Fusion, Modeling and Simulation. She is the Editor-in Chief to International Journal of Advanced Intelligence Paradigms (IJAIP), member in Editorial Board member of several national and international journals and is evaluator expert for national and international projects. G.Chitra is a Part time Research Scholar at Anna University, Chennai working for her Ph.D degree under the supervision of Prof.Dr.V.Yegnanarayanan, the main author of this paper. She is also working as a full time faculty member in the capacity as Assistant Professor in the Department of Mathematics at Velammal Engineering College, Chennai.

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Introduction

Graph Theory is a useful tool when try to analyze and understand large and complex networks. Modeling network traffic and finding the shape of the internet are some of the practical applications of graph theory on networking [?]. In mathematics, graph theory is defined as the study of the properties of graphs. Common graph theory problems include route problems, finding network flow, graph coloring, finding subgraphs etc. Graph Theory can be used to determine optimal placement of intrusion detection agents within a network. See [?]. A graph is a mathematical object interpreted as a set of vertices and a set of edges that join some or all of the vertices. If two vertices in a graph are connected by an edge, they are said to be adjacent, else is said to be nonadjacent. The domination number of a graph is a notable graph invariant. The idea of domination is based on sets of vertices that ”are near” all the vertices of a graph. There are numerous graphical invariants defined for graphs.

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Graph Domination Numbers

We denote the vertex set of a graph by V (G), or simply V. The number of edges incident to a vertex v is the degree of the vertex deg(v) and two vertices are adjacent if they are incident to the same edge. Definition 1. A vertex set D is a dominating set if for every vertex u ∈ V − D, u is adjacent to at least one

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vertex in D. The domination number γ(G) is the minimum cardinality among all dominating sets in G. [?] Definition 2. A set D is a total dominating set if for every vertex u ∈ V , u is adjacent to at least one vertex in D. The total domination number γt (G) is the minimum cardinality among all total dominating sets in G.[?] Definition 3. For a vertex v, the neighborhood N (v) is the set of all vertices adjacent to v and the closed neighborhood of a vertex u is N [u] = N (u) ∪ u.[?] Definition 4. A dominating set D is called a locating-dominating set if for any two vertices v, w ∈ V − D, N (v) ∩ D ̸= N (w) ∩ D. The locating domination number of a graph G is the minimum cardinality among all locating dominating sets in G and is denoted by γL (G).[?] Definition 5. A dominating set D is called a differentiating dominating set if for any two vertices v, w ∈ V, N [v] ∩ D ̸= N [w] ∩ D. The differentiating domination number of a graph G is the least vertex set size among all differentiating dominating sets in G and is denoted by γD (G).[?] Definition 6. The global alliance number γa (G) of a graph G is the minimum cardinality among all global alliances of G, where a set D is a global alliance if D is a dominating set and for each u ∈ D, the number of ”allies” it has in D are at least as many as it has in V - D. In other words, D is a dominating set and for each vertex u ∈ D, it is true that |N [u] ∩ D| ≥ |N (u) ∩ (V − D)|.[?] The adjacency matrix A = A(G) and the degree matrix D∗ = D∗ (G) are the square matrices that contain information about the internal connectivity of vertices in G. Definition 7. We define an adjacency matrix as one in which Ai,j = 1 if and only if vi and vj are adjacent ∗ = deg(vi ) if i = j and 0 otherwise.[?] and o otherwise; Di,j

Definition 8. The Laplacian matrix L = Li,j (G) is the square matrix defined by L = D∗ −A; Li,j = deg(vi ) if i = j,−1 if i ̸= j and (vi , vj ) ∈ E(G), 0 otherwise.[?] Observation 1. The eigenvalues of the Laplacian matrix of a graph is the graph’s spectrum. The eigenvalues are relation with the density distribution of the set of edges. The second least eigenvalue,denoted by λ2 is the best measure of the graph’s connectivity among all of the eigenvalues. Higher values for λ2 correspond to vertices of high degree that are in close proximity whereas small values for λ2 point out to a dispersed edge set.[?] In next theorem, Theorem 1, is about the locating domination numbers of paths and cycles. Theorem 1: For n ∈ Z, with n ≥ 4, γL (G) = ⌊n/2⌋ if G ∼ = Pn or Cn . Proof. Let G ∼ = Pn where Pn = v1 v2 ...vn−1 vn . If n = 2m, then pick v2i for 1 ≤ i ≤ m as the m elements of a set D. As (v2i−1 , v2i ) ∈ E(Pn ) for 1 ≤ i ≤ m, we deduce that D is a dominating set. We

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now claim that for any two vertices vr , vs ∈ V − D, N (vr ) ∩ D ̸= N (vs ) ∩ D. Clearly vr = v2i1 −1 and vs = v2i2 −1 for some i1 , i2 ∈ {1, 2, ..., m}. Note that either i1 > i2 or i1 < i2 . Without loss of generality assume that i1 < i2 . Then as (v2i1 −2 , v2i1 −1 ), (v2i1 −1 , v2i1 ), (v2i2 −2 , v2i2 −1 ), (v2i2 −1 , v2i2 ) ∈ E(G) we infer that N (v2i1 −1 ) = {v2i1 −2 , v2i1 } and N (v2i2 −1 ) = {v2i2 −2 , v2i2 }. It is now easy to check that N (v2i1 −1 ) ∩ D = {v2i1 −2 , v2i1 } ̸= {v2i2 −2 , v2i2 } = N (v2i2 −1 ) ∩ D. Therefore γL (G) ≥ m. Next we claim that γL (G) ≤ m. Suppose that γL (G) < m. Then γL (G) ≤ m − 1. If all the (m − 1) elements of D are internal vertices of G, then as the degree of each such vertex is 2, they can dominate at most 2m − 2 vertices, a contradiction. If (m − 1) elements of D has one external vertex of G, then they can dominate among them at most 2(m − 2) + 1 = 2m − 3 vertices , a contradiction. If (m − 1) elements of D has two external vertices of G, then they can dominate among them at most 2(m − 3) + 2 = 2m − 4 vertices , a contradiction. Therefore γL (G) ≤ m and hence γL (G) = m. The case of n = 2m+1, can be dealt with on similar lines. Next let G ∼ = Cn . Let Cn = v1 v2 ...vn−1 vn v1 . If n = 2m + 1, then, pick v2i−1 for 1 ≤ i ≤ m as the m elements of a set D. As (v2i−1 , v2i ) ∈ E(G) for 1 ≤ i ≤ m , and (v2m+1 , v1 ) ∈ E(G) we deduce that D is a dominating set. We now claim that for any two vr , vs ∈ V − D, N (vr ) ∩ D ̸= N (vs ) ∩ D. Clearly vr = v2i1 and vs = v2i2 for some i1 , i2 ∈ {1, 2, ..., m}. Note that either i1 > i2 or i1 < i2 . Without loss of generality assume that i1 < i2 . Note that N (v2i1 ) = {v2i1 −1 , v2i1 +1 } and N (v2i2 ) = {v2i2 −1 , v2i2 +1 } and hence N (v2i1 ) ∩ D = {v2i1 −1 , v2i1 +1 } = ̸ {v2i2 −1 , v2i2 +1 } = N (v2i2 ) ∩ D. Therefore γL (G) ≥ m. Suppose that γL (G) ≤ m − 1. As G is a cycle , deg(vi ) = 2 for all 1 ≤ i ≤ 2m + 1. Therefore if a dominating set D has only m − 1 elements, then it can dominate at most 2m − 2 elements , a contradiction as |V (G)| = 2m + 1. Therefore γL (G) = ⌊(2m + 1)/2⌋ = m. The case that n = 2m can be proved on similar lines and hence γL (G) = ⌊n/2⌋ if G ∼ = Pn or Cn . In next theorem, Theorem 2, is about the differentiating domination number of paths. Theorem 2: Let n ∈ Z + , with n ≥ 5. Then γD (Pn ) = n − 2. Proof. Let G ∼ = Pn where Pn = v1 v2 ...vn−1 vn . If n = 2m + 1, then pick all the internal vertices of G as the elements of a set D. Then as (v1 , v2 ), (v2m+1 , v2m ) ∈ E(G), we deduce that D is a dominating set of G. We now claim that for any two vertices vr , vs ∈ V (G), N [vr ] ∩ D ̸= N [vs ] ∩ D. If both vr , vs ∈ D then as r > s or r < s, without loss of generality assume that r < s. Now N [vr ] ∩ D = {vr−1 , vr , vr+1 } ̸= {vs−1 , vs , vs+1 } = N [vs ] ∩ D, we conclude that D is differentiating dominating set and γD (G) ≥ n − 2. If vr , vs ∈ V − D with r < s, then clearly vr = v1 and vs = v2m+1 . So N [v1 ] ∩ D =

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{v1 , v2 } ̸= {v2m , v2m+1 } ∩ D and we conclude that D is a differentiating dominating set and γD (G) ≥ n − 2. If vr ∈ D and vs ∈ V − D, with r < s, then N [vr ] ∩ D = {vr−1 , vr , vr+1 } ∩ D = {vr−1 , vr , vr+1 } if vr is an internal vertex of G and N [vr ] ∩ D = {v1 , v2 } ∩ D = {v2 } or {v2m , v2m+1 } ∩ D = {v2m } if vr is an external vertex of G and N [vs ] ∩ D = {vs−1 , vs , vs+1 } ∩ D = {vs−1 , vs , vs+1 } if vs is an internal vertex of G and N [vs ] ∩ D = {v1 , v2 } ∩ D = {v2 } or {v2m , v2m+1 } ∩ D = {v2m } if vs is an external vertex of G and hence N [vr ]∩D = {vr−1 , vr , vr+1 } = ̸ {vs−1 , vs , vs+1 } = N [vs ]∩D; N [vr ]∩D = {vr−1 , vr , vr+1 } ̸= {v2 } = N [vs ]∩D; N [vr ] ∩ D = {vr−1 , vr , vr+1 } ̸= {v2m } = N [vs ] ∩ D; N [vr ] ∩ D = {v2 } ̸= {vs−1 , vs , vs+1 } = N [vs ] ∩ D; N [vr ] ∩ D = {v2m } ̸= {vs−1 , vs , vs+1 } = N [vs ] ∩ D; N [vr ] ∩ D = {v2 } ̸= {v2m } = N [vs ] ∩ D. So in all the above cases, we infer that D is a differentiating dominating set and γD (G) ≤ n − 2. Suppose that γD (G) ≤ n − 3, then such a dominating set D must be of one of the following types. ” Type 1. The three vertices that D missed out are all internal vertices ” Type 2. Out of the three vertices that D missed out, two are internal vertices and one is an external vertex. ” Type 3. Out of the three vertices that D missed out, one is an internal vertex and the other two areexternalvertices. ” If D is of Type 1, then choose the 2m-1  ways. Observe that N [v1 ] ∩ D = {v1 } and internal vertices as v2 , v3 and v4 out of the total  3 N [v2 ] ∩ D = {v1 } where D = {v1 , v5 , ..., v2m , v2m+1 },a contradiction. ” If D is of Type 2, then choose the internal vertices as v2 , v3 out of the total 

2m − 1

 ways and v2m+1 as the external vertex. Observe that 2 N [v1 ] ∩ D = {v1 } and N [v2 ] ∩ D = {v1 } where D = {v1 , v4 , ..., v2m−1 , v2m }, a contradiction. If D is of Type 3, then choose both v1 , v2m+1 , the external vertices and for the choice of an internal vertex, select out of the (2m − 1) choices, the vertex v2 . Then D = {v3 , v4 , , v2m−1 , v2m }. But then as v1 and v2 are both in V \ D, D cannot be a dominating set, a contradiction. Therefore γD (G) = n − 2 as the case that n = 2m can be proved on similar lines. In next theorem, Theorem 3, is about the differentiating domination number of cycles. Theorem 3: For n ∈ Z, with n ≥ 5, γD (Cn ) = n − 2 if n is odd and n/2 if n is even. Proof. Let G ∼ = Cn where Cn = v1 v2 ..., vn−1 vn v1 . If n = 2m + 1, then pick either the first (n − 2) vertices or the last (n−2) vertices as the elements of a set D. In the case of former, as (v2m−1 , v2m ), (v2m+1 , v1 ) ∈ E(G), we deduce that D is a dominating set of G. We now claim that for any two vertices vr , vs ∈ V (G), N [vr ]∩D ̸= N [vs ] ∩ D. Note that either r > s or r < s. Without loss of generality assume that r < s. Let vr , vs ∈ D. Then N [vr ] ∩ D = {vr−1 , vr , vr+1 } ̸= {vs−1 , vs , vs+1 } = N [vs ] ∩ D. So γD (Cn ) ≥ n − 2. If vr , vs ∈ V − D then clearly vr = v2m and vs = v2m1 . So N [v2m ] ∩ D = {v2m1 } = ̸ {v1 } = N [v2m+1 ] ∩ D. Hence γD (Cn ) ≥ n − 2. If

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γr ∈ D and γs ∈ V − D then two cases arise. Case (i) vr ∈ D and vs = v2m ;Case (ii) vr ∈ D and vs = v2m+1 . In case (i). we have N [v2m ] ∩ D = {v2m−1 }, N [vr ] ∩ D = {vr−1 , vr , vr+1 } ̸= {v2m−1 } = N [vs ] ∩ D. And in case (ii) we have N [v2m+1 ] ∩ D = {v1 } ̸= {vr , vr−1 , vr+1 } = N [vr ] ∩ D. So γD (G) ≥ n − 2. Next we claim that γD (G) ≤ n − 2. Suppose that γD (G) = n − 3. If V \ D contains three consecutive elements, say v2m−1 , v2m and v2m+1 , then in that case D will not be a dominating set, a contradiction. The case of latter can be dealt with on similar lines. So γD (Cn ) = n − 2, if n is odd. If n = 2m then select either D = {v2i |1 ≤ i ≤ m} or D = {v2i−1 |1 ≤ i ≤ m} . In the case of former, as v2i−1 , v2i for 1 ≤ i ≤ m, we deduce that D is a dominating set. Next we claim that for any two vr , vs ∈ V (G), N [vr ] ∩ D ̸= N [vs ] ∩ D. Suppose that vr , vs ∈ D, then vr = v2i and vs = v2j for some i, j ∈ {1, 2, ..., m}. Now N [v2i ] ∩ D = {v2i } ̸= {v2j } = N [v2j ] ∩ D. If vr , vs ∈ V \ D then vr = v2i1 −1 and vs = v2i2 −1 for some i1 , i2 ∈ {1, 2, ., m}. Now N [v2i1 −1 ] ∩ D = {v2i1 , v2i1 −2 } ̸= {v2i2 } = N [v2i2 −1 ] ∩ D. If vr ∈ V − D and vs ∈ D, then vr = v2i1 −1 for some i ∈ {1, 2, ., m} and vs = v2i for some i ∈ {1, 2, ., m}. Now N [v2i1 −1 ] ∩ D = {v2i1 −2 , v2i1 } ̸= {v2i } = N [v2i ] ∩ D . Therefore we infer that γD (G) ≥ m. We now claim that γD (G) ≤ m. Suppose that γD (G) = m − 1. If a dominating set D has only m − 1 elements, then it can dominate at most 2m − 2 elements , a contradiction as |V (G)| = 2m, a contradiction. The latter case can also be dealt with on similar lines as above. Therefore γD (G) =

n 2,

if n is even.

In next theorem, Theorem 4, is about the global alliance domination number of paths and cycles. Theorem 4: Let G = Cn or Pn , for n ≥ 4. Then γa (G) = n − 2. Proof. Let G = Pn with Pn = v1 v2 ..., vn−1 vn . Pick D = {v2 , ....., vn−1 }. As (v1 , v2 ), (vn−1 , vn ) ∈ E(G), D is a dominating set. Let v ∈ D be any arbitrary element, then v = vs for some 2 ≤ s ≤ (n − 1). Clearly N [v] = {vs−1 , vs , vs+1 } and N [v] ∩ D = {vs−1 , vs , vs+1 }, N [v] ∩ V \ D = {∅}. So |N [v] ∩ D| = 3 ≥ |N [v] ∩ V \ D| = 0 if vs ̸= v2 , vn−1 . Now N [v2 ] = {v1 , v2 , v3 } implies N [v2 ] ∩ D = {v2 , v3 }, N [v2 ] ∩ V \ D = {v1 }. Hence |N [v2 ] ∩ D| = 2 > |N [v2 ] ∩ V \ D| = 1. Similarly it can be seen for v = vn−1 . Therefore γa (G) ≥ n − 2. Next we claim that γa (G) = n − 2. Suppose not and γa (G) = n − 3. Let vp , vq , vr are the elements in V \ D. Here three cases arise. Case 1. All of them are internal vertices of G. Suppose not and choose vp = v2 , vq = v3 , vr = v5 . Then N [4] = {v3 , v4 , v5 }. So N [4] ∩ D = {v4 }; N [4] ∩ V \ D = {v3 , v5 }. That is |N [v4 ] ∩ D| = 1 < |N [v4 ] ∩ V \ D| = 2, a contradiction. Case 2. vp and vq are internal vertices of G. Suppose not and choose vp = v3 . vq = v4 and vs = v1 ,then N [v2 ] = {v1 , v2 , v3 }. So N [2] ∩ D = {v2 }; N [2] ∩ V \ D = {v1 , v3 }. That is |N [v2 ] ∩ D| = 1 < |N [v2 ] ∩ V \ D| = 2, a contradiction. Case 3. vp alone is an internal vertex of G. Suppose not and choose vp = v3 , vq = v1 , vs = vn , then N [v2 ] = {v1 , v2 , v3 }.

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So N [2] ∩ D = {v2 }; N [2] ∩ V \ D = {v1 , v3 }. That is |N [v2 ] ∩ D| = 1 < |N [v2 ] ∩ V \ D| = 2, a contradiction. Therefore γa (G) = n − 2.

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Laplacian Matrix and Eigen values

The Laplacian matrix has a long history. Because of its importance in various theories of physical and chemical nature, the Laplacian matrix’s spectrum is widely studied than the adjacency spectrum. In [?] Alon used the smallest positive eigenvalue of the Laplacian matrix to study the expander and magnifying coefficients of graphs. The gap between the second and the first eigen values is an extremely important parameter in many branches of mathematics. If the graph is connected, then the largest eigen value of the adjacency matrix as well as the smallest eigen value of the Laplacian matrix have multiplicity 1. We can expect that the gap between this and the nearest eigen value is related to some kind of connectivity measure of the graph. From the matrix - tree theorem, λ2 (G), the second smallest eigen value of an Laplacian matrix of G is positive if and only if G is connected. This observation motivated Fiedler to define the algebraic connectivity of G by λ2 (G). It is considered as a quantitative measure for connectivity. In [?] Fiedler has proved the following significant result about the upper and lower bounds for the second smallest eigen value in terms of vertex connectivity and edge connectivity of a graph. Theorem 5 ([?]): Let G be a simple graph of order n other than the complete graph with vertex connectivity κ(G) and edge connectivity κ′ (G). Then 2κ′ (G)(1 − cos(π/n)) ≤ λ2 (G) ≤ κ(G) ≤ κ′ (G). In [?] Kirkland et.al obtained the necessary and sufficient conditions for λ2 (G) = κ(G). Theorem 6 ([?]): Let G be a simple connected graph of order n other than a complete graph. Then λ2 (G) = κ(G) if and only if G = G1 ∨ G2 , where G1 is a disconnected graph of order n − κ(G) and G2 is a graph of order κ(G) with λ2 (G) ≥ 2κ(G) − n. In [?], Lu et.al gave an upper bound for λ2 in terms of the domination number. Theorem 7 ([?]): Let G be a simple graph of order n ≥ 2. Then λ2 (G) ≤ (n(n − 2γ(G) + 1))/n − γ(G) with equality if and only if G ∼ = K2,2 .

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In [?], Nikiforov gave another upper bound. Theorem 8 ([?]): Let G be a simple graph of order n other than a complete graph. Then λ2 (G) ≤ n − γ(G). In [?], Lu et.al found the following result that links λ2 and the diameter of a graph G. Theorem 9 ([?]): Let G be a simple connected graph of order n with m edges and diameter diam(G). Then λ2 (G) ≥ 2n/(2 + n(n − 1)(diam(G)) − 2m(diam(G))). In [?], Zhang obtained the following results. Theorem 10 ([?]): Let T be a tree of order n with independence number α(T ). If T ∼ ̸= K1,n−1 or Tn,n−2 , then λ2 (T ) ≤

√ 3− 5 2

with equality if and only if T is Tn,α(T ) . Let RV be the set of functions from V to R, RV = {f : V → R}. RV becomes a real vector space ∑ f (u)g(u). The corresponding norm in R is of dimension n endowed with the inner product ⟨f, g⟩ = u∈V √ ∑ 2 ∥ f ∥= ⟨f, f ⟩ = ( f (u))1/2 . The matrix L = L(G) acts on R and performing the role of a linear u∈V

operator. Matrix vector multiplication rule determines its action. So g = Lf is the function defined by ∑ L(uv)f (v), u ∈ V . There is a canonical quadratic form associated with the formula g(u) = (Lf )(u) = v∈V

L = L(G). The multiplicity of the value 0 as an eigen value of L(G) is equal to the number of connected components of G. To see this, let K be a connected component of G. Let χ(H) ∈ RV be the characteristic function of V (H). That is, χ(H)(v) is equal to 1 if and only if v ∈ V (H). It is easy to note that L(G)f (H) =0. Since the characteristic functions of different connected components are linearly independent, the multiplicity of the eigen value 0 is atleast the number of connected components of G. In fact, λ1 (G) = 0 is a simple eigen value of L(G) if and only if G is connected.

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Some parameters based on Domination Measures

One of the novelty of the current paper is the combination of the domination measures mentioned in Section 3 with the newly introduced parameters: P1 ,P2 and P3 . Define P1 = (γ + γt + γa )/n and P2 = (γL + γD )/n and P3 = (γL + γD + nλ2 ). Further we calculate, compare and analyze these parameters for all connected graphs on four vertices, see Table 1. Several Observations are following. λ1 (G) = 0 for G = Gi , i = 1, 3 to

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6, is a simple root of multiplicity one and all G′i s are connected. We notice from the Table-1 that when the graph is dense and regular to the maximum permissible complete graph the gap λ2 − λ1 is large. When the graph is sparse and tight with more pendant vertices the value of (λ2 − λ1 ) is 1 and especially for the tight graph P4 , the value of (λ2 − λ1 ) is the least. Interestingly for two graphs with equal number of edges, viz, G3 and G6 the gap (λ2 − λ1 ) is different and it is less for a non-regular graph and more for a regular graph. In G′i s, i = 1 to 6, except for G4 all other graphs are non-complete, simple and connected. Out of these only for G3 , G5 and G6 we have λ2 = κ. See Table-1. Moreover we observe that G3 = H1 ∨ H2 with H1 = K2 ∪ K1 ; H2 = K1 , where H1 is a disconnected graph of order n − κ(G3 ) = 4 − 2 = 2 and H2 is a graph of order κ(G3 )(= 2) with λ2 (G3 ) = 2 ≥ 2κ(G3 ) − n = 2 ∗ 2 − 4 = 0. These observations very well satisfies the conditions given in Theorem 6. In terms of domination number, we have the following upper bound [?], λ2 (G) ≤ n − γ(G) by Theorem 8. From the Table-1, we see that λ2 (G1 ) = 0.5858 ≤ 4 − 2 = 2; λ2 (G2 ) = 1 ≤ 4 − 1 = 3; λ2 (G3 ) = 2 ≤ 4 − 2 = 2;λ2 (G4 ) = 4 ≤ 4 − 1 = 2; λ2 (G5 ) = 2 ≤ 4 − 2 = 2; λ2 (G6 ) = 2 ≤ 4 − 2 = 2. So all G3 , G5 and G6 are the extremal graphs for the graph equation λ2 (G) = n − γ(G). In terms of diameter of a graph, we have the following lower bound [?]. λ2 (G) ≥ 2n/(2+n(n−1)(diam(G))− 2m(diam(G))) by Theorem 9. From the table, we see that λ2 (G1 ) = 0.5858 ≥ (2 ∗ 4)/(2 + 4(4 − 1)(3 − (6 ∗ 3))) = −0.044943; λ2 (G2 ) = 1 ≥ (2 ∗ 4)/(2 + 4(4 − 1)(2 − (2 ∗ 4 ∗ 2))) = −0.048193; λ2 (G3 ) = 2 ≥ (2∗4)/(2+4(4−1)(2−(2∗5∗2))) = −0.037383; λ2 (G4 ) = 4 ≥ (2∗4)/(2+4(4−1)(1−(2∗6∗1))) = −0.061538; λ2 (G5 ) = 2 ≥ (2 ∗ 4)/(2 + 4(4 − 1)(2 − (2 ∗ 5 ∗ 2))) = −0.037383; λ2 (G6 ) = 2 ≥ (2 ∗ 4)/(2 + 4(4 − 1)(2 − (2 ∗ 4 ∗ 2))) = −0.048193; Interestingly none of these graphs are the extremal graphs for the graph equation

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λ2 (G) = 2n/(2 + n(n − 1)(diam(G)) − 2m(diam(G))). Theorem 10, gives another useful upper bound for λ2 (G). We notice that only the graph G2 ∼ = K1,3 satisfies the inequality 0 = λ2 (G2 ) ≤

√ 3− 5 2

= 0.38; Theorem 10 provides a fact that λ2 (G) ≤ (n(n − 2γ(G) + 1)/(n −

γ(G)). Now from Table-1 we deduce that except for the complete graph G5 = K4 all other graphs complies with the above inequality. That is, λ2 (G1 ) = 05858 ≤ (4(4−2∗2+1)/4−2) = 2; λ2 (G2 ) = 1 ≤ (4(4−2∗1+1)/4−1) = 4; λ2 (G3 ) = 2 ≤ (4(4 − 2 ∗ 2 + 1)/4 − 2) = 2; λ2 (G4 ) = 4  (4(4 − 2 ∗ 2 + 1)/4 − 2) = 2; λ2 (G5 ) = 2 ≤ (4(4 − 2 ∗ 2 + 1)/4 − 2) = 2; λ2 (G6 ) = 2 ≤ (4(4 − 2 ∗ 2 + 1)/4 − 2) = 2. Moreover the graphs G3 , G4 , G5 and G6 are the extremal graphs for the graph equation λ2 (G) = (n(n − 2γ(G) + 1)/(n − γ(G)). Next Theorem 5 reveals a nice relation connecting λ2 with the measures of vertex connectivity and the edge connectivity. Notice that: 2 ∗ 1 ∗ (1 − cos(π/4)) = 2 ∗ 029289 ≤ λ2 (G1 ) = 0.5858 ≤ 1 = κ(G1 ) ≤ 1 = κ′ (G1 ); 2 ∗ 1 ∗ (1 − cos(π/4)) = 2 ∗ 029289 ≤ λ2 (G2 ) = 1 ≤ 1 = κ(G2 ) ≤ 1 = κ′ (G2 ); 2 ∗ 2 ∗ (1 − cos(π/4)) = 4 ∗ 029289 ≤ λ2 (G3 ) = 2 ≤ 2 = κ(G4 ) ≤ 2 = κ′ (G4 ); 2 ∗ 2 ∗ (1 − cos(π/4)) = 4 ∗ 029289 ≤ λ2 (G5 ) = 2 ≤ 2 = κ(G5 ) ≤ 2 = κ′ (G6 ); 2 ∗ 2 ∗ (1 − cos(π/4)) = 4 ∗ 029289 ≤ λ2 (G6 ) = 2 ≤ 1 = κ(G6 ) ≤ 2 = κ′ (G7 ). Here G′i s for i = 1, 2, 3, 4, 6 all satisfies the above relation. The graphs G2 , G3 , G5 are the extremal graphs for the graph equation λ2 (G) = κ(G) = κ′ (G). All G′i s for i = 1, 2, 3, 4, 6 are not extremal graphs for the graph equation 2 ∗ κ′ (G) ∗ (1 − cos(π/n)) = λ2 (G). By ACSGOO4, we mean all connected simple graphs of order 4. Figure 1 exhibits P1 and P2 data for ACSGOO4. Figure 2 shows the P3 and P1 − P2 comparison data for ACSGOO4. Figure 3 shows the P1 − P3 and P2 −P3 comparison data for ACSGOO4. Figure 4 shows the P1 −P2 −P3 comparison data for ACSGOO4.

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In what follows we briefly discuss about some applications of domination numbers in WSN’s.

5

Applications of Domination Numbers

Wireless Sensor Networks (WSN) are used, for instance, in safety and military applications for the purpose of monitoring and tracking geographic boundaries. They are also used for automating manufacturing processes, for monitoring the structures of building, and in monitoring the environmental systems for forests, oceans and precision farming. WSNs have substantial data acquisition and data processing capabilities and for this reason are deployed densely throughout the area where they will monitor specific phenomena [?] and [?]. A WSN can be modeled as a graph G = (V, E), where V is the set of sensor nodes, and E is the set of wireless links. For Example one can deem that each node in V is associated to its Euclidean coordinates and a link between any pair of nodes u and v exists (i.e., (u, v) ∈ E) if and only if the Euclidean distance between the pair of nodes is smaller than or equal to their transmission range (assumed to be the same for all nodes in V ). Connectivity, primarily a graph-theoretic concept, helps define the fault tolerance of WSNs in the sense that it enables the sensors to communicate with each other so that the sink can be reached by their sensed data. Further the sensing coverage, a vital architectorial feature of WSN plays an important role in meeting particular applications. for instance, to extract relevant data reliably that concern a sensed field. The concepts of Sensing coverage and network connectivity are not orthogonal. However it has been established that connectivity depends on coverage. A subset D of sensor nodes V of a WSN graph G = (V, E) is said to be a dominating set if every node in V − D is connected to some node in D. The minimum number of nodes in a dominating set of G is called domination number of G.

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Domination set problem: Given a graph G and a positive integer k, Is Domination number of G is less than or equal to k ? That is, γ(G) ≤ k. In [?], Cockayne et. al. had shown the NP Completeness of the domination problem. Here we outline a procedure to find the cluster heads in a cluster group using the idea of domination set. There are a number of graph theoretic approaches available in the literature for clustering in WSNs and there are a wide range of such algorithms, each may be suitable for a different cross-layer design objective.

6

Conclusions

In this we have computed certain interesting variations of domination numbers of popular graphs like paths and cycles and probed the scope of their practical applications. Research in the field of Wireless sensor networks is growing rapidly and achieving tangible results that apply to real life scenarios. Research attention needs to be directed to the following areas [?] and [?]:Tolerate the physical security inadequacy;Optimizing the security infrastructure in terms of resources (energy and computation;Detecting and reacting to DoS attacks;Raise social privacy related issues;Manage and protect the mobile nodes and base stations;Multiple base stations has to be governed in a secured way with delegations of privileges.

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