Bounds on inverse sum indeg index of graph operations

<p>Cut vertices are often used as a measure of nodes’ importance within a network. These are nodes whose failure disconnects a connected graph. Let <span class="math inline">\(N(G)\)</span> be the number of connected induced subgraphs of a graph <span class="math inline">\(G\)</span>. In this work, we investigate the maximum of <span class="math inline">\(N(G)\)</span> where <span class="math inline">\(G\)</span> is a unicyclic graph with <span class="math inline">\(n\)</span> nodes of which <span class="math inline">\(c\)</span> are cut vertices. For all valid <span class="math inline">\(n,c\)</span>, we give a full description of those maximal (that maximise <span class="math inline">\(N(.)\)</span>) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of <span class="math inline">\(n,c\)</span>, however, there is a unique maximal unicyclic graph with <span class="math inline">\(n\)</span> nodes and <span class="math inline">\(c\)</span> cut vertices. In particular, the well-known negative correlation between the number of connected induced subgraphs of trees and the Wiener index (sum of distances) fails for unicyclic graphs with <span class="math inline">\(n\)</span> nodes and <span class="math inline">\(c\)</span> cut vertices: for instance, the maximal unicyclic graph with <span class="math inline">\(n=3,4\mod 5\)</span> nodes and <span class="math inline">\(c=n-5>3\)</span> cut vertices is different from the unique graph that was shown by Tan et al. [<span><em>The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices</em></span>. J. Appl. Math. Comput., 55:1–24, 2017] to minimise the Wiener index. Our main characterisation of maximal unicyclic graphs with respect to the number of connected induced subgraphs also applies to unicyclic graphs with <span class="math inline">\(n\)</span> nodes, <span class="math inline">\(c\)</span> cut vertices and girth at most <span class="math inline">\(g>3\)</span>, since it is shown that the girth of every maximal graph with <span class="math inline">\(n\)</span> nodes and <span class="math inline">\(c\)</span> cut vertices cannot exceed <span class="math inline">\(4\)</span>.</p>

<p>The Euler-Sombor index <span class="math inline">\(EU\)</span> is a vertex-degree-based graph invariant, defined as the sum over all pairs of adjacent vertices <span class="math inline">\(u,v\)</span> of the underlying graph, of the terms <span class="math inline">\(\sqrt{d_u^2+d_v^2+d_u\,d_v}\)</span>, where <span class="math inline">\(d_u\)</span> and <span class="math inline">\(d_v\)</span> are the degrees of the vertices <span class="math inline">\(u\)</span> and <span class="math inline">\(v\)</span>, respectively. For a real number <span class="math inline">\(\lambda\)</span>, a variable version of <span class="math inline">\(EU\)</span> is constructed, denoted by <span class="math inline">\(EU(\lambda)\)</span>, defined via <span class="math inline">\(\sqrt{d_u^2+d_v^2+\lambda\,d_u\,d_v}\)</span>. Its special cases for <span class="math inline">\(\lambda=2,\,-2,\,0\)</span>, and 1 are, respectively, the first Zagreb, Albertson, Sombor, and the ordinary Euler-Sombor indices. The basic properties of <span class="math inline">\(EU(\lambda)\)</span> are determined, including a method for its approximate calculation and bounds in terms of minimum degree, maximum degree, order and size for several graph products. It is shown how to find values of <span class="math inline">\(\lambda\)</span> for which <span class="math inline">\(EU(\lambda)\)</span> is optimal with regard to predicting molecular properties.</p>

<p>For a connected graph <span class="math inline">\(G=(V,E)\)</span> of order at least two, a <span class="math inline">\(u-v\)</span> chordless path in <span class="math inline">\(G\)</span> is a <span class="math inline">\(monophonic\)</span> <span class="math inline">\(path\)</span>. The <em>edge monophonic closed interval</em> <span class="math inline">\(I_{em}[u,v]\)</span> consists of all the edges lying on some <span class="math inline">\(u-v\)</span> monophonic path. For <span class="math inline">\(S'\subseteq V(G),\)</span> the set <span class="math inline">\(I_{em}[S']\)</span> is the union of all sets <span class="math inline">\(I_{em}[u,v]\)</span> for <span class="math inline">\(u,v\in S'.\)</span> A set <span class="math inline">\(S'\)</span> of vertices in <span class="math inline">\(G\)</span> is called an <span class="math inline">\(edge\)</span> <span class="math inline">\(monophonic\)</span> <span class="math inline">\(set\)</span> of <span class="math inline">\(G\)</span> if <span class="math inline">\(I_{em}[S']=E(G).\)</span> The <em>edge monophonic number <span class="math inline">\({m_1}(G)\)</span> of G</em> is the minimum cardinality of its edge monophonic sets of <span class="math inline">\(G\)</span>. In this paper the monophonic number and the edge monophonic number of corona product graphs are obtained. Exact values are determined for several classes of corona product graphs.</p>

<p>Let <span class="math inline">\(A\)</span> be a commutative ring with nonzero identity and <span class="math inline">\(n\geq 2\)</span> be a positive integer. With the ring <span class="math inline">\(R=A\times\cdots\times A\)</span> (<span class="math inline">\(n\)</span> times), one can associate graphs <span class="math inline">\(TD(R)\)</span> and <span class="math inline">\(ZD(R)\)</span> respectively called the total dot product graph and the zero-divisor dot product graph of <span class="math inline">\(R\)</span>. In this paper, we study some topologicaal properties of these two dot product graphs of <span class="math inline">\(R.\)</span> In particular, it is shown that, the zero-divisor dot product graph <span class="math inline">\(ZD(R)\)</span> is a projective graph if and only if <span class="math inline">\(R\)</span> is isomorphic to <span class="math inline">\(\frac{Z_2\left[x\right]}{\left\langle x^2+x+1\right\rangle}\times\frac{Z_2\left[x\right]}{\left\langle x^2+x+1\right\rangle}.\)</span> Moreover, we prove that no total dot product graph can be projective. With these observations, we classify all commutative rings for which dot product graphs <span class="math inline">\(ZD(R)\)</span> and <span class="math inline">\(TD(R)\)</span> have crosscap two.</p>

<p>For a graph, a smallest-last (SL) ordering is formed by iteratively deleting a vertex with smallest degree, then reversing the resulting list. The SL algorithm applies greedy coloring to a SL ordering. For a given vertex coloring algorithm, a graph is hard-to-color (HC) if every implementation of the algorithm results in a nonoptimal coloring. In 1997, Kubale et al. showed that <span class="math inline">\(C_{8}^{2}\)</span> is the unique smallest HC graph for the SL algorithm. Extending this, we show that for <span class="math inline">\(k\geq4\)</span>, <span class="math inline">\(k+4\)</span> is the smallest order of a HC graph for the SL algorithm with <span class="math inline">\(\chi\left(G\right)=k\)</span>. We also present a HC graph for the SL algorithm with <span class="math inline">\(\chi\left(G\right)=3\)</span> that has order 10.</p>

LetG=G1×G2×⋯×Gmbe the strong product of simple, finite connected graphs, and letϕ:ℕ⟶0,∞be an increasing function. We consider the action of generalized maximal operatorMGϕonℓpspaces. We determine the exact value ofℓp-quasi-norm ofMGϕfor the case whenGis strong product of complete graphs, where0<p≤1. However, lower and upper bounds ofℓp-norm have been determined when1<p<∞. Finally, we computed the lower and upper bounds ofMGϕpwhenGis strong product of arbitrary graphs, where0<p≤1.

<div class="htmlview paragraph">This paper presents results of a study conducted to apply and evaluate the In-Vehicle Information System (IVIS) DEMAnD Model developed recently by the Virginia Polytechnic University's Center for Transportation Research for the Federal Highway Administration. This software-based model allows vehicle design engineers to predict the effects an in-vehicle information system might have on driver performance. The model was exercised under nine different driver attention task levels ranging from simple, such as glancing into a side view mirror, to complex, such as operating an in-vehicle navigation system. The nine driver tasks were evaluated using three different vehicle configurations and two levels of driver-roadway complexity. In addition, real-world information on driver visual performance was also collected during four different tasks for comparison with model predictions of these same functions. The comparison of model prediction for maximum number of glances, total task performance time, and a model-rating feature called <i>figure of demand</i> for each of the tasks indicated:</div> <div class="htmlview paragraph"> <ol class="list nostyle"> <li class="list-item"><span class="li-label">1</span><div class="htmlview paragraph">Driver task performance behavior is influenced substantially more by differences in the level of driver/roadway/traffic combinations, than by differences in test vehicle configurations.</div></li> <li class="list-item"><span class="li-label">2</span><div class="htmlview paragraph">The driver performance data collected under actual driving conditions compared very well with the model predictions.</div></li> <li class="list-item"><span class="li-label">3</span><div class="htmlview paragraph">Overall, the IVIS DEMAnD model appears to be a good early attempt at modeling the effect on driver performance of in-vehicle IT (Information Technology) systems in general.</div></li> </ol> </div>

Strong products of Kneser graphs

This paper provides new observations on the Lovász -function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed in terms of the second-largest and smallest eigenvalues of the adjacency matrix of the regular graph, together with sufficient conditions for equalities (the upper bound is due to Lovász, followed by a new sufficient condition for its tightness). These results are shown to be useful in many ways, leading to the determination of the exact value of the Shannon capacity of various graphs, eigenvalue inequalities, and bounds on the clique and chromatic numbers of graphs. Since the Lovász -function factorizes for the strong product of graphs, the results are also particularly useful for parameters of strong products or strong powers of graphs. Bounds on the smallest and second-largest eigenvalues of strong products of regular graphs are consequently derived, expressed as functions of the Lovász -function (or the smallest eigenvalue) of each factor. The resulting lower bound on the second-largest eigenvalue of a k-fold strong power of a regular graph is compared to the Alon–Boppana bound; under a certain condition, the new bound is superior in its exponential growth rate (in k). Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and the Lovász -function of each factor. The utility of these bounds is exemplified, leading in some cases to an exact determination of the chromatic numbers of strong products or strong powers of graphs. The present research paper is aimed to have tutorial value as well.

A variation of the channel-assignment problem is naturally modeled by L(2,1)-labelings of graphs. An L(2,1)-labeling of a graph G is an assignment of labels from {0,1,...,/spl lambda/} to the vertices of G such that vertices at distance two get different labels and adjacent vertices get labels that are at least two apart and the /spl lambda/-number /spl lambda/(G) of G is the minimum value /spl lambda/ such that G admits an L(2,1)-labeling. The /spl Delta//sup 2/-conjecture asserts that for any graph G its /spl lambda/-number is at most the square of its largest degree. In this paper it is shown that the conjecture holds for graphs that are direct or strong products of nontrivial graphs. Explicit labelings of such graphs are also constructed.

Let G be a simple connected graph of order n, m edges, maximum degree Δ 1 and minimum degree δ. Li et al. (Appl. Math. Lett. 23:286-290, 2010) gave an upper bound on number of spanning trees of a graph in terms of n, m, Δ 1 and δ: t ( G ) ≤ δ ( 2 m − Δ 1 − δ − 1 n − 3 ) n − 3 . The equality holds if and only if G ≅ K 1 , n − 1 , G ≅ K n , G ≅ K 1 ∨ ( K 1 ∪ K n − 2 ) or G ≅ K n − e , where e is any edge of K n . Unfortunately, this upper bound is erroneous. In particular, we show that this upper bound is not true for complete graph K n . In this paper we obtain some upper bounds on the number of spanning trees of graph G in terms of its structural parameters such as the number of vertices (n), the number of edges (m), maximum degree ( Δ 1 ), second maximum degree ( Δ 2 ), minimum degree (δ), independence number (α), clique number (ω). Moreover, we give the Nordhaus-Gaddum-type result for number of spanning trees. MSC:05C50, 15A18.

Graph-theoretic properties of certain proximity graphs defined on planar point sets are investigated. We first consider some of the most common proximity graphs of the family of the Delaunay graph, and study their number of edges, minimum and maximum degree, clique number, and chromatic number. In the second part of the paper we focus on the higher order versions of some of these graphs and give bounds on the same parameters.

Homothetic polygons and beyond: Maximal cliques in intersection graphs

Distance and Eccentric sequences to bound the Wiener index, Hosoya polynomial and the average eccentricity in the strong products of graphs

Maximal Clique Enumeration (MCE) is a long standing problem in database community. Though it is extensively studied, almost all researches focus on calculating maximal cliques as a one-time effort. MCE on dynamic graph has been rarely discussed so far, the only work on this topic is to maintain maximal cliques with graph evolving. The key within this problem is to find maximal cliques that contains vertices incident to the inserted edge when edge insertion happens. Up to O(W2) candidates are generated in prior method based on Cartesian product, the overall complexity is O(W2n2) where n, W represents the number of vertices and maximal cliques on the graph. Besides, maximality verification of candidate is conducted frequently by global search. Change of maximal clique induced by graph's updating presents some localities. We propose novel local construction strategy to generate candidates based on linear scan, number of candidates is reduced to O(W), the overall complexity is then reduced to O(Wn2). Furthermore, we present heuristics to reduce the cost incurred by maximality verification. Theoretical analysis and experiments on real graphs indicate that our proposals are effective and efficient.