scholarly journals Clusters in a Multigraph with Elevated Density

10.37236/4847 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Mark K. Goldberg
Keyword(s):  
Lower Bounds ◽  
Vertex Degree ◽  
Chromatic Index ◽  
Gamma Delta ◽  
Dense Sets

In this paper, we prove that in a multigraph whose density $\Gamma$ exceeds the maxiimum vertex degree $\Delta$, the collection of minimal clusters (maximally dense sets of vertices) is cycle-free. We also prove that for multigraphs with $\Gamma\gt\Delta+1$, the size of any cluster is bounded from the above by $(\Gamma-3)/(\Gamma-\Delta-1)$. Finally, we show that two well-known lower bounds for the chromatic index of a multigraph are equal.

scholarly journals Wiener–Hosoya Matrix of Connected Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 359
Author(s):  
Hassan Ibrahim ◽  
Reza Sharafdini ◽  
Tamás Réti ◽  
Abolape Akwu
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Wiener Index ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Connected Graphs ◽  
Matrix Description ◽  
Vertex Set

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

scholarly journals On some mathematical properties of the general zeroth-order Randić coindex of graphs

2020 ◽  
Vol 12 (2) ◽  
pp. 75-82
Author(s):  
I. Milovanović ◽  
M. Matejić ◽  
E. Milovanović ◽  
A. Ali
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Connected Graph ◽  
Degree Sequence ◽  
Tree Structure ◽  
Vertex Degree ◽  
Arbitrary Real Number ◽  
Zeroth Order ◽  
Mathematical Properties ◽  
Simple Connected Graph

Let G = (V,E), V = {v1, v2,..., vn}, be a simple connected graph of order n, size m with vertex degree sequence ∆ = d1 ≥ d2 ≥ ··· ≥ dn = d > 0, di = d(vi). Denote by G a complement of G. If vertices vi and v j are adjacent in G, we write i ~ j, otherwise we write i j. The general zeroth-order Randic coindex of ' G is defined as 0Ra(G) = ∑i j (d a-1 i + d a-1 j ) = ∑ n i=1 (n-1-di)d a-1 i , where a is an arbitrary real number. Similarly, general zerothorder Randic coindex of ' G is defined as 0Ra(G) = ∑ n i=1 di(n-1-di) a-1 . New lower bounds for 0Ra(G) and 0Ra(G) are obtained. A case when G has a tree structure is also covered.

scholarly journals Asymptotic Lower Bounds on Circular Chromatic Index of Snarks

10.37236/2388 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Martin Mačaj ◽  
Ján Mazák
Keyword(s):  
Lower Bounds ◽  
Alternative Proof ◽  
Cubic Graphs ◽  
Chromatic Index ◽  
Infinite Class ◽  
Cubic Graph ◽  
Asymptotically Optimal

We prove that the circular chromatic index of a cubic graph $G$ with $2k$ vertices and chromatic index $4$ is at least $3+2/k$. This bound is (asymptotically) optimal for an infinite class of cubic graphs containing bridges. We also show that the constant $2$ in the above bound can be increased for graphs with larger girth or higher connectivity. In particular, if $G$ has girth at least $5$, its circular chromatic index is at least $3+2.5/k$. Our method gives an alternative proof that the circular chromatic index of the generalised type 1 Blanuša snark $B_m^1$ is $3+2/3m$.

scholarly journals Bounds for Distinguishing Invariants of Infinite Graphs

10.37236/6362 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Wilfried Imrich ◽  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Mohammad Hadi Shekarriz
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Infinite Graphs ◽  
Chromatic Index ◽  
Distinguishing Number ◽  
Minimum Number ◽  
Edge Colourings ◽  
Delta G ◽  
Vertex Colouring

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that $\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree $\Delta(G)$, and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$, where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result $\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of conjectures are formulated.

scholarly journals Sharp Upper and Lower Bounds of VDB Topological Indices of Digraphs

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1903
Author(s):  
Juan Monsalve ◽  
Juan Rada
Keyword(s):  
Lower Bounds ◽  
Topological Index ◽  
General Setting ◽  
Upper Bounds ◽  
Topological Indices ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Extremal Values

A vertex-degree-based (VDB, for short) topological index φ induced by the numbers φij was recently defined for a digraph D, as φD=12∑uvφdu+dv−, where du+ denotes the out-degree of the vertex u,dv− denotes the in-degree of the vertex v, and the sum runs over the set of arcs uv of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over Dn, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over OTn and OG, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively.

Generalized edge-colorings of weighted graphs

2016 ◽  
Vol 08 (01) ◽  
pp. 1650015
Author(s):  
Yuji Obata ◽  
Takao Nishizeki
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Edge Coloring ◽  
Efficient Algorithms ◽  
Upper And Lower Bounds ◽  
Weighted Graphs ◽  
Chromatic Index ◽  
Edge Colorings ◽  
Integer Weight

Let [Formula: see text] be a graph with a positive integer weight [Formula: see text] for each vertex [Formula: see text]. One wishes to assign each edge [Formula: see text] of [Formula: see text] a positive integer [Formula: see text] as a color so that [Formula: see text] for any vertex [Formula: see text] and any two edges [Formula: see text] and [Formula: see text] incident to [Formula: see text]. Such an assignment [Formula: see text] is called an [Formula: see text]-edge-coloring of [Formula: see text], and the maximum integer assigned to edges is called the span of [Formula: see text]. The [Formula: see text]-chromatic index of [Formula: see text] is the minimum span over all [Formula: see text]-edge-colorings of [Formula: see text]. In the paper, we present various upper and lower bounds on the [Formula: see text]-chromatic index, and obtain three efficient algorithms to find an [Formula: see text]-edge-coloring of a given graph. One of them finds an [Formula: see text]-edge-coloring with span smaller than twice the [Formula: see text]-chromatic index.

scholarly journals Bounds on the weighted vertex PI index of cacti graphs

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 5977-5989
Author(s):  
Gang Ma ◽  
Qiuju Bian ◽  
Jianfeng Wang
Keyword(s):  
Lower Bounds ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Pi Index ◽  
Weighted Vertex

The weighted vertex PI index of a graph G is defined by PIw(G)=?e=uv?E(G) (dG(u)+dG(v))(nu(e|G) + nv(e|G)) where dG(u) denotes the vertex degree of u and nu(e|G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. A graph is a cactus if it is connected and all its blocks are either edges or cycles. In this paper, we give the upper and lower bounds on the weighted vertex PI index of cacti with n vertices and s cycles, and completely characterize the corresponding extremal graphs.

scholarly journals Bounds and extremal graphs of second reformulated Zagreb index for graphs with cyclomatic number at most three

2021 ◽  
Vol 49 (1) ◽  
Author(s):  
Abhay Rajpoot ◽  
◽  
Lavanya Selvaganesh ◽  
Keyword(s):  
Lower Bounds ◽  
Topological Indices ◽  
Simple Approach ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Cyclomatic Number ◽  
Tricyclic Graphs

Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

Effects of Increased Dietary Protein and Energy on Composition and Functional Capacities of Blood Mononuclear Cells from Vaccinated, Neonatal Calves

2005 ◽  
Vol 75 (5) ◽  
pp. 357-368 ◽  
Author(s):  
Foote ◽  
Nonnecke† ◽  
Waters ◽  
Palmer ◽  
Beitz ◽  
...  
Keyword(s):  
Weight Gain ◽  
B Cell ◽  
Dietary Protein ◽  
Crude Protein ◽  
Interleukin 2 ◽  
Milk Replacer ◽  
Standard Diet ◽  
Gamma Delta ◽  
Ifn Gamma ◽  
Functional Capacities

Effects of increased protein and energy provided by an intensified milk replacer on the antigen-specific, cell-mediated immune response of the neonatal calf were examined. Calves were fed a standard (0.45 kg/day of a 20% crude protein, 20% fat milk replacer; n = 11) or intensified (1.14 kg/day of a 28% crude protein, 20% fat milk replacer; n = 11) diet from 0 to 6 weeks of age. All calves were vaccinated with Mycobacterium bovis bacillus Calmette-Guerin (BCG) at 1 week of age. The daily weight gain of intensified-diet calves (0.62 kg/day) was greater than the weight gain of standard-diet calves (0.29 kg/day). Liver, kidney, heart, thymus, and subcervical lymph nodes from intensified-diet calves were heavier than the same organs from standard-diet calves. Flow cytometric analysis of peripheral blood mononuclear cell (PBMC) populations indicated that CD4+ cells, gamma delta TCR+ cells, and monocyte percentages, although unaffected by diet during the first 5 weeks of the study, were higher in intensified-diet calves at week 6. The decline in gamma deltad TCR+ cell percentages and increase in B cell percentages with increasing age seen in all calves are characteristic of the maturing immune system of the calf. CD8+ T cell or B cell percentages were not affected by diet. In intensified-diet calves, percentages of CD4+ expressing interleukin-2 receptor increased and percentages of gamma delta TCR+ cells expressing interleukin-2 receptor decreased with time. The same populations in standard-diet calves did not change with time. Percentages of CD4+ and CD8+ T cells, and B cells expressing MHC class II antigen, were unaffected by diet or age. Although mitogen-induced interferon (IFN)-gamma and nitric oxide (NO) secretion increased with age for all calves, PBMC from intensified-diet calves produced less IFN-gamma and more NO than did cells from standard-diet calves at week 6 of the study. Antigen-induced secretion of IFN-gamma and NO also increased with age but was unaffected by diet. Antigen-elicited delayed-type hypersensitivity was unaffected by diet, suggesting increased dietary protein and energy did not alter adaptive immunity in vivo. Overall, these results suggest that feeding calves a commercially available, intensified milk replacer affects minimally the composition and functional capacities of PBMC populations. Additional research is necessary to determine whether these subtle effects influence the calf’s susceptibility to infectious disease.

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