n . In this paper, we consider several natural vertex weights of G and obtain some inequalities between the ordinary and Laplacian energies of G with correspond- ing vertex weights. Finally, we apply our results to the molecular **graph** of toroidal fullerenes (or achiral polyhex nanotorus).

Abstract In this paper, we introduce a new type of graph energy called the non-common-neighborhood energy ENCN G , NCN-energy for some standard graphs is obtained and an upper bound fo[r]

A large number of results on the **graph** **energy** have been reported, see for instance [2, 8, 10, 13]. Motivated by the success of the theory of **graph** **energy**, other different **energy** like quantities have been proposed and studied by different researchers. Let D(G) = [d ij ] be the diagonal matrix associated with the **graph** G, where

[25] Y. Z. Song, P. Arbelaez, P. Hall, C. Li, A. Balikai, Finding semantic struc- tures in image hierarchies using Laplacian **graph** **energy**, in: K. Daniilidis, P. Maragos, N. Paragios (Eds.), Computer Vision – CECV 2010 (European Conference on Computer Vision, 2010), Part IV , Springer, Berlin, 2010, pp. 694–707.

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For details on the mathematical aspects of theory of **graph** **energy** see the papers [4, 5] and the references cited there in. The basic properties including various upper and lower bounds for **energy** of a **graph** have been established in [7, 8] and it has found remarkable chemical applications in the molecular orbital theory of conjugated molecules [2, 6].

its adjacency matrix. The studies on the **graph** **energy** can be seen in papers [5, 6]. For a de- tailed survey on applications to the **graph** **energy**, see [2–4, 7]. For the ordinary **energy**, the best known bounds are the Koolen and Moulton upper bound [9, 10] and the McClelland lower bound [12].

Theories on the mathematical concepts of **graph** **energy** can be seen in the reviews[3], papers [4] [5] [6] and the references cited there in. For various upper and lower bounds for **energy** of a **graph** can be found in papers[7] [8] and it was observed that **graph** **energy** has chemical applications in the molecular orbital theory of conjugated mo- How to cite this paper: Kanna, M.R.R. and

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J. Liu and B. Liu defined the Seidel **energy** **graph** in generalization for Laplacian **energy** and analyze. For the Seidel **energy** bound using the rank of the sediel matrix and extended the concept of **energy** to Hermite matrix. Willem H. Harmers in Seidel switching and **graph** **energy** investigates that Seidel switching changes the spectrum, but not the **energy** and present an infinite family of example with maximal **energy**. In this chapter the seidel matrix [15] which is the generalization of the adjacency matrix is analyzed for sharp bound of the Seidel **energy** for the connected and disconnected **graph**. In this paper we find few bounds which relate domination number of book **graph** , and Seidel **energy** of book **graph** and rank of the incident matrix of the book **graph** and pose some open problems for further research. The motivation for this paper is to establish a link between there parameters. This is the chapter motivated from the Paper [1]. CLEMENTS BRAND, NORBERT SEITER, Eigen values and determination in **graph**, Mathematic Slovaca, Vol.46 (1996). No. 1.1,33 – 39, where the authors had found the relation between largest eigen value of the Laplacian matrix and Domination number.

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Finally you've got to steer the beam so it smashes into something. The two smashing methods are fixed target and colliding beams. For fixed target, you slam a beam of particles into some stationary chunk of matter. The advantages of the fixed target method are: (a) it's easier to aim the beam at a large non-moving target, (b) lots and lots of protons are available in chunks of matter, and (c) most all the products of the particle interaction will be moving in a very definite forward direction, so you can easily build detectors in the forward region to detect them. The disadvantages are that chunks of matter pretty much only provide protons and neutrons to smash into--the electrons aren't concentrated enough to be consequential--and there are certainly no anti-protons or positrons (anti-electrons). Also, a target which is just sitting there does not bring any kinetic **energy** into the equation. Since you are typically trying to convert **energy** into massive particles (via E = mc 2 ), more **energy** is better.

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In the nonvoting based process a node with a valid certificate selects a node as a malicious aggressor to revoke its certificate. J. clulow et al [9] planned the suburbanized suicide-based approach “suicide for the good strategy, certificate is revoked through only 1 accusation. This will at constant time revokes certificates of every accused node associated exculpatory node to urge obviate AN aggressor from the network; the exculpatory node has to sacrifice itself. Even though the process gradually reduces the time necessary to eject a node as well as communication overhead, but this suicidal approach doesn't take into thought to differentiate falsely suspect nodes from real malicious attackers. Park et al. [10] planned a cluster based certificate revocation theme; where nodes endure self-organized in the direction of create clusters. A positive certification authority do liable in the direction of hold the individual and suspect node at intervals the warning list (WL), blacklist (BL) and manages management messages correspondingly. And in addition it deals with the matter of false accusation the malicious aggressor node certificate is revoked by any solitary neighbouring node. Thus takes short time to complete certificate revocation methodology. [12]In this paper, author given associate in nursing **energy**-efficient distributed clump process for ad-hoc networks. The approach is hybrid: cluster heads area unit haphazardly hand-picked supported their on the market **energy** and nodes connect with clusters specified communication price is decreased. During this paper, author proposes a replacement **energy** economical approach for clump nodes in unintentional device networks. Author presents a protocol, HEED that sporadically selects cluster heads per a hybrid of their residual **energy** and node proximity towards its neighbours or node degree to boost network life

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Let G be a **graph**, 𝑥𝑥 𝑖𝑖 (𝐺𝐺) be the number of vertices of degree 𝑖𝑖, 𝑥𝑥(𝐺𝐺) be the order of G, and 𝑦𝑦 𝑖𝑖𝑖𝑖 (𝐺𝐺) be the number of edges that link vertices of degree i and j. A chemical **graph** is a **graph** in that no vertex has degree greater than 4. A common linked **graph** with x vertices and x + 1 edges is called a bicyclic **graph**. Assume that 𝑣𝑣 1 is a vertex of 𝑇𝑇 𝑒𝑒 and 𝑣𝑣 𝑝𝑝 is a vertex of 𝑇𝑇 𝑟𝑟 . Combing 𝑣𝑣 1 and 𝑣𝑣 𝑝𝑝 by a path 𝑣𝑣 1 , 𝑣𝑣 2,,,,,, 𝑣𝑣 𝑝𝑝 of length p−1, where p ≥ 1 (p = 1 means determine 𝑣𝑣 1 with 𝑣𝑣 𝑝𝑝 ), the answering **graph**, described by D(e, p, r) described in Figure 1, is called an ∞-**graph**. Let 𝑃𝑃 𝑒𝑒 +1 , 𝑃𝑃 𝑝𝑝+1 𝑎𝑎𝑎𝑎𝑑𝑑 𝑃𝑃 𝑟𝑟+1

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In this paper, the Randi´c Estrada index of a **graph** is introduced. Also the Randi´c **energy** and the Randi´c Estrada index are studied. In section 2, a sharper upper bound and a new lower bound for the Randi´c **energy** are obtained. In section 3, some bounds for the Randi´c Estrada index involving Randi´c index, Randi´c **energy** and some other **graph** invariants are also put forward.

Let the **graph** G(V;E) where V be vertices and E edge. **Graph** theory involves lot of applications such as geometry, algebra, number theory, topology, operations research, and optimization and computer science. In 1967, most **graph** labeling methods trace their origin introduced by Rosa [11] and Graham and Sloane [5] introduced harmonious graceful **graph** by additive bases in 1980. Gallery proof of lebeling results given by Gallian[6] in aDynamic Survey of **Graph** Labeling. Pradhan and Kumar [10] proved that graphs obtained by adding a pendent vertex of hair cycle ʘ are graceful if n≡0(mod4m).They further provide a rule for determining the missing numbers in the graceful labeling of ʘ and of the **graph** obtained by adding pendent edges to each pendent vertex of ʘ . Abhyanker[2] proved that the **graph** obtained by deleting the branch of length 1 from an olive tree with 2n branches and identifying the root of the edge deleted tree with a vertex of a cycle of the form is graceful. In 1985 Koh and Yap[8] generalized this by defining a cycle with a -chord to be a cycle with the path joining two nonconsecutive vertices of the cycle.

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The increasing number of phosphorus atoms in the nanotubes relative increase in absolute total **energy** is the electron **energy**. Other results of this study was to obtain structural relationship with the electron **energy** using the QSAR And electron **energy** levels for the four other nanotubes were predicted on the basis of this method.

In 1965 Fuzzy relations on a set was first defined by Zadeh [14].The nature of values discussed in **energy** of **graph** is never an odd integer [3] and **energy** of a **graph** is never the square root of an odd integer [13].Study on **energy** of different graphs such as regular, non-regular, circulate and random graphs [4, 5, 6] by I.Gutman and shao.In [1] Anjali Narayanan and Sunil Mathew introduced the concept of the **energy** of fuzzy **graph**.

4. AGREEMENTS BETWEEN E AND RE Within various classes of molecular graphs, a reasonably good (increasing) linear correlation between Randić **energy** and total π-electron **energy** could be established. As characteristic examples, we present here such correlations for chemical trees (Figure 1) and catacondensed benzenoid hydrocarbons (Figure 2). From Figures 1 and 2 (as well as from the analogous computational results not presented here), we conclude that the main structural factors that influence the total π-electron **energy** also influence (in the same sense) the Randić **energy**. Recall that in the case of chemical trees, .

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Inspired by the initial success of GSP in many fields [15], [16], [17], in this paper, we propose a GSP-based NALM ap- proach by extending [17] to perform low-complexity multiclass classification of the acquired active power readings without the need for event detection to detect appliance changing states, e.g., switching on/off. We index the acquired power signal by a directed **graph** where each vertex corresponds to a load sample and the weights of the edges connecting the vertices reflect the degree of similarity between the nodes. Then, we define an optimization problem that contains as the regularization term the total **graph** variation, that is, we apply regularization on the constructed **graph** signal to find a signal with minimum variation. Our experimental results using publicly available REDD dataset [18] demonstrate high potentials of the proposed approach.

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In this paper, we construct some new classes of reciprocal graphs and an upperbound for their **energy** is obtained. Pairs of equienergetic reciprocal graphs on n ≡ 0 mod ( 12 ) and n ≡ 0 mod ( 16 ) are constructed. The Wiener indices of some classes of reciprocal graphs are also obtained. These results are not found so far in literature.

This graph invariant is very closely related to a chemical quantity known as the total π-electron energy of conjugated hydrocarbon molecules, see [1,2,4,5].. The energy of all isome[r]

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From **Graph** 1, it is observed that pull-down time of domestic refrigerator by using refrigerant R-436a is higher as compared to that of refrigerant R-134a by about 25%. From **Graph** 5, it is observed that **energy** consumed for pull-down period of domestic refrigerator by using refrigerant R-436a is lower as compared to that of refrigerant R-134a by about 23.6% which is reflected by better COP.