Middle Graph Research Articles

On Co-complete k–partite Graph Valued Functions

A co-complete k -partite graph G = (V1,V2, … ,Vk, E), k ≥ 2 is a graph with the smallest number k of disjoint parts in which any pair of vertices in the same part is at a distance two. The number of parts in co-complete k -partite graph G is denoted by k (G). In this paper, we investigate k() , k(L(G)), k(M(G)) and k(T(G)), where , L(G) , M(G) and T(G) are complement graph, line graph, middle graph and total graph, respectively of some standard graphs. We also discuss the relationship between them. Each set in the partition has subpartitions such that each set in the subpartition induces K1 or any two vertices in this subpartition are at distance two and this result has significance in providing a stable network.

Proof of the middle levels conjecture

Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length 2 n + 1 that have exactly n or n + 1 entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ in exactly one bit. The middle levels conjecture asserts that this graph has a Hamilton cycle for every n ⩾ 1 . This conjecture originated probably with Havel, Buck and Wiedemann, but has also been attributed to Dejter, Erdös, Trotter and various others, and despite considerable efforts, it resisted all attacks during the last 30 years. In this paper, we prove the middle levels conjecture. In fact, we construct 2 2 Ω ( n ) different Hamilton cycles in the middle layer graph, which is best possible.

Boundary Domination of Line and Middle Graph of Wheel Graph Families

Let G = (V;E) be a connected graph. A subset S of V (G) is called a boundary dominating set if every vertex of V S is boundary dominated by some vertex of S. The minimum taken over all boundary dominating sets of a graphG is called the boundary domination number ofG and is denoted by b(G). We define the boundary domatic number in graphs. Exact values of of Wheel Graph Families are obtained and some other interesting results are established.

Rice cropping density and intensity lessened in southeast China during the twenty-first century.

Accurate and updated time series maps of paddy rice distribution and planting intensity will greatly improve our knowledge. Unfortunately, spatiotemporal explicit information on rice fields is relatively limited, and considerable uncertainties still exist as regards to its inter-annual variations in China. In this study, an improved rice mapping methodology was proposed through combined consideration of vegetation phenology and surface moisture variations from different seasonal rice. This method was applied to southeast China based on 500 m 8 day composite Moderate Resolution Imaging Spectroradiometer (MODIS) Enhance Vegetation Indices with two bands (EVI2) during the period 2001-2013. Its efficiency was validated with 763 ground survey sites, with an overall accuracy of 95.02 % and the kappa index of 0.9217. Spatiotemporal analysis indicated that rice cropping density and intensity lessened in southeast China during the period 2001-2013. Particularly, the paddy rice-planted areas reduced by 30.09 %, changing from 231,005 to 161,484 km(2). Among them, the planted areas of double rice decreased by 49.34 %, changing from 34,215 to 17,335 km(2). Therefore, averaged rice cropping intensity in southeast China decreased from 1.148 to 1.107. The primary dynamic patterns were from single rice or a rotation of rice plus other crops to non-rice (93,386 km(2)) and double rice to non-double rice (24,132 km(2)). When analyzed at provincial and altitudinal gradient levels, it was obvious that areas with greater rice cropping density or intensity were associated with more remarkable reductions. Graphical abstract The left graph shows that the rice cropping density lessened in Hubei, Hunan, Guangdong, Jiangxi, Anhui, Jiangsu, Henan provinces and other three provincial-level administrative units (Zhejiang, Fujian and Shanghai) from 2001 to 2013. The middle graph indicates the movement of gravity center as well as the variations in the total planted areas of single rice, rice plus others and double rice. The right graph denotes that the rice cropping intensity decreased in each provincial-level administrative unit from 2001 to 2013.

On Middle Cube Graphs

We study a family of graphs related to the $n$-cube. The middle cube graph of parameter k is the subgraph of $Q_{2k-1}$ induced by the set of vertices whose binary representation has either $k-1$ or $k$ number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors).

A note on the middle levels problem

The middle levels problem consists in determining a hamiltonian cycle in the bipartite Kneser graph B(2k+1,k), also known as the middle levels graph and denoted by Bk. Previously, it was proved that a particular hamiltonian path in a reduced graph of Bk implies a hamiltonian cycle in Bk and a hamiltonian path in the Kneser graph K(2k+1,k). We show that the existence of such a particular hamiltonian path in a reduced graph of K(2k+3,k) implies a hamiltonian path in K(2k+3,k) for k≡1 or 2(mod3). Moreover, we utilize properties from the middle levels graphs to improve a known algorithm speeding up the search for such a particular hamiltonian path in the reduced graph of Bk.

Some Path Related 4-Cordial Graphs

In this paper, we discuss 4-cordial labeling of some path related graphs. We prove that middle graph, total graph and splitting graph of the path are 4-cordial. In addition to this we prove that $P_n^2 $ and triangular snakes are 4-cordial.

Promiscuous T-cell responses to drugs and drug-haptens

Promiscuous T-cell responses to drugs and drug-haptens

A Note on b-Coloring of Fan Graphs

A proper coloring of the graph assigns colors to the vertices, edges, or both so that proximal elements are assigned distinct colors. Concepts and questions of graph coloring arise naturally from practical problems and have found applications in many areas, including Information Theory and most notably Theoretical Computer Science. A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color class. The b-chromatic number of a graph G, denoted by φ(G), is the maximal integer k such that G may have a b-coloring with k colors. In this paper, the authors find the b-chromatic number of middle and total graph of fan graph.

Zeta functions and complexities of middle graphs of semiregular bipartite graphs

We give a determinant expression for the zeta function of the middle graph of a semiregular bipartite graph G and express the complexity of the middle graph of G by using the complexity of G. As a corollary, we obtain an explicit formula for the complexity of the middle graph of a complete bipartite graph.

On b-chromatic number of sun let graph and wheel graph families

A proper coloring of the graph assigns colors to the vertices, edges, or both so that proximal elements are assigned distinct colors. Concepts and questions of graph coloring arise naturally from practical problems and have found applications in many areas, including Information Theory and most notably Theoretical Computer Science. A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color class. The b-chromatic number of a graph G, denoted by φ(G), is the maximal integer k such that G may have a b-coloring with k colors. In this paper, we obtain the b-chromatic number for the sun let graph Sn, line graph of sun let graph L(Sn), middle graph of sun let graph M(Sn), total graph of sun let graph T(Sn), middle graph of wheel graph M(Wn) and the total graph of wheel graph T(Wn).

On $$b$$ b -chromatic number with other types of chromatic numbers on double star graphs

In this paper, we obtain the $$b$$ -chromatic number for the central graph, middle graph, total graph and line graph of double star graph $$K_{1,n,n}$$ denoted by $$C(K_{1,n,n}), M(K_{1,n,n}), T(K_{1,n,n})$$ and $$L(K_{1,n,n})$$ respectively. We find a relationship between the $$b$$ -chromatic number and three other coloring parameters, the equitable chromatic number, harmonious chromatic number and the achromatic number. Also we conjecture that the $$b$$ -chromatic number is bounded by the harmonious chromatic number.

The b-chromatic number of bistar graph

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer ’(G)) for which a graph has a b-coloring with ’(G) colors. In this paper, we established the b-chromatic number of the central graph, middle graph, and total graph of the bistar graph Bm;n. Mathematics Subject Classication: 05C15, 05C76

Dominator Coloring Number of Middle Graph

Given a graph G, the dominator coloring problem seeks a proper coloring of G with the additional property that every vertex in the graph dominates an entire color class. In this paper, as an extension of Dominator coloring some standard results for the middle graph of path and cycle has been discussed. Let G = (V(G), E(G)) be a graph with n = |V(G)| and m = |E(G)|. For any vertex v V(G), the open neighborhood of v is the set N (v) = {u| uv  E (G)} and the closed neighborhood is the set N(v) = N(v)  v. Similarly, for any set SV (G), N(S) =  vs N (v) -S and N(S) = N(S)  S. A set S is a dominating set if N(S)

The 2-Pebbling Property of the Middle Graph of Fan Graphs

A pebbling move on a graphGconsists of taking two pebbles off one vertex and placing one pebble on an adjacent vertex. The pebbling number of a connected graphG, denoted byf(G), is the leastnsuch that any distribution ofnpebbles onGallows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. This paper determines the pebbling numbers and the 2-pebbling property of the middle graph of fan graphs.

Some New Results on Distance k-Domination in Graphs

We determine the distance k-domination number for the total graph, shadow graph, and middle graph of path Pn.

ON VERTEX DISJOINT CYCLES OF MIDDLE AND TOTAL GRAPH OF $K_n$ AND $K_{m,n}$

Tulgeity �(G) of a graph G is the maximum number of vertex disjoint cycles contained in G. In this paper the basic results on tulgeity of a graph have been reviewed and the formula for the tulgeity of the middle and total graph of complete graph and complete bigraph are derived. Also an upper bound for the tulgeity of middle graph of any graph is presented and the graph for which the tulgeity attains its upper bound has been classified. Cycle is a significant feature of a graph. The analysis of cycle plays an important role in the design and development of some graph model. Several problems of finding shortest path, shortest spanning trees, least cost Hamiltonian cycles, etc., of a graph have been studied. We discuss the problems of finding the maximum number of vertex disjoint cycles. We consider finite, simple, undirected graph G(V (G),E(G)) where V (G) and E(G) represent vertex set and edge set of G respectively. p and q denote the

Disruption of transfer entropy and inter-hemispheric brain functional connectivity in patients with disorder of consciousness

Recent studies suggest that the functional organization of brain networks is altered in patients with severe disorders of consciousness (DOC), including coma [1]. A better characterization of these large-scale disturbances of brain function is fundamental for the understanding and eventual prediction of coma outcome. In this study we apply information theory tools and correlation measures to explore the brain functional connectivity (FC) in functional magnetic resonance imaging (fMRI) data from patients with DOC. In particular we are interested in the functional coherence within and between brain hemispheres as a measure of the integrity of brain function. The data analyzed comprises resting state fMRI data from 5 patients with traumatic DOC in two sessions, one early after the traumatic event and the second later in a stable clinical phase spanning several months. As a control we used similar data obtained from 17 healthy subjects. FC was assessed among time series of the blood oxygenated level dependent (BOLD) signal from 90 regions of interest (ROI) obtained using the Automated Anatomical Labelling (AAL) atlas [2]. These time series were used to estimate FC using two approaches: Partial Linear Correlation (PLC) and Transfer Entropy (TE) [3]. The major findings are summarized in Figure ​Figure1;1; the PLC analysis shows a significant reduction of the inter-hemispheric FC in DOC patients compared to controls. Regarding TE measures, we have found a strong reduction in the total TE of the DOC patients vs controls; mean values of 59.17 bits for DOC vs 123.97 bits for controls, whilst the two groups have a similar standard deviation (62.41 for DOC vs 76.69 for controls). Furthermore, the results based on TE seem to indicate that the inter-hemispheric FC asymmetry reported by the PLC in Figure ​Figure11 might be corresponded with an asymmetry between the TE travelling from to vs the one from to along the same ROI in the AAL. Notice that TE (but not PLC) is a non-symmetric measure. Interestingly, some of the DOC patients presented a FC pattern closer to the healthy subjects, for instance patient 3 who in the second session (i.e., 3' in Figure ​Figure1)1) was fully conscious. These findings might have clinical value for the diagnosis and design of recovery plans for patients with DOC. Figure 1 For each subject it is shown the PLC summed over the 45 regions of the left hemisphere (bottom graph), the right hemisphere (middle graph) or between all right-left homologous ROIs in the AAL (top graph). Numbers on x-axis are indicating the different ...

Radio Labeling For Some Cycle Related Graphs

Let $G=(V(G),E(G))$ be a connected graph and let $d(u,v)$ denote the distance between any two vertices in $G$. The maximum distance between any pair of vertices is called the diameter of $G$ denoted by $diam(G)$. A radio labeling( or multilevel distance labeling) for $G$ is an injective function $f:V(G) \longrightarrow N\cup\{0\}$ such that for any vertices $u$ and $v$, $|f(u)-f(v)| \geq diam(G)-d(u,v)+1$. The span of $f$ is the largest number in $f(V)$. The radio number of $G$, denoted by $r_n(G)$ is the minimum span of a radio labeling of $G$. In this paper we determine upper bounds of radio numbers for cycle with chords and $n/2$-petal graph. Further the radio number is completely determined for the split graph and middle graph of cycle $C_{n}$.

Construction of 2-factors in the middle layer of the discrete cube

Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length 2n+1 that have exactly n or n+1 entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ in exactly one bit. In this work we present an inductive construction of a large family of 2-factors in the middle layer graph for all n⩾1. We also investigate how the choice of certain parameters used in the construction affects the number and lengths of the cycles in the resulting 2-factor.