Super Edge-connectivity and Zeroth-order General Randić Index for −1 ≤ α < 0

The thesis consists of three chapters. The first chapter introduces the basic notions of graph theory and defines vertex-reconstruction and edge-reconstruction problem. The second chapter and third chapter are devoted to the edge-reconstruction of bi-degreed graphs and bipartite graphs respectively. A bi-degreed graph G is a graph with two degrees d > δ. By elementary arguments we can assume d = δ + 1 and there are at least two vertices of degree δ. Call vertices of degree d "big" vertex and degree δ "small" vertex. Define "symmetric" path of length p Sp to be one with both ends small vertices and all other internal vertices big vertices; define "asymmetric" path of length p Ap to be one with one end a small vertex and all others big vertices. If s(G) is the minimum distance between two small vertices in G, we can show that s(G) is "independent" of G (i.e. it is edge-reconstructable), and that G has at most one nonisomorphic edge-reconstruction H. From this, the concept of "forced move" posed by Dr. Swart is obvious. Using the principle of forced move (and sometimes also "forced edge" posed by Dr. Swart as well), it's easy to derive a few interesting properties, like say G is edge-reconstructable if s(G) is even or if two Ss(G)'s intersect at an internal vertex, etc. Write s for s(G). When s is odd, consider the concept of s - n-chain, which is n Ss's following from end to end. We can show first s - 3-chain and then s - 2-chain cannot exist. Hence Ss's are disjoint. Think of Ss's as "lines" in some geometry. Define two more "distance" functions s1 and s2 such that s1 "represents" the distance from a point to a line and s2 means the distance between to "skew" lines. With the aid of forced move principle again, we can at last prove every bi-degreed graph with at least four edges is edge-reconstructable. A bipartite graph G is a graph whose vertex set V is the disjoint union of two sets v1 and v2 such that every edge joins v1 and v2. By elementary reduction we can assume G to be connected. We define special chains inductively so that it starts at a vertex of minimum degree and always goes to a neighbor or minimum degree. Special chains will be the main tool to prove edge-reconstructability. By G's finiteness, we note they will "terminate" somewhere, and we have three types of termination for them. Let condition A•s be that degree sequence of special chain is edge-reconstructable, condition Bi's be that number of special chains is edge-reconstructable(and some more general variations); condition P's be that the "last vertices" of two special chains be not adjacent; we can prove that all A, Bi and P's should hold inductively in an interlocked way. (This is a big task). Then condition P's can be used to prove G's edge-reconstructability for all three types of termination. We can then prove every bipartite graph with at least four edges is edge-reconstructable.

Let D be a finite and simple digraph with vertex set V(D). The minimum degree δ of a digraph D is defined as the minimum value of its out-degrees and its in-degrees. If D is a digraph with minimum degree δ and edge-connectivity λ, then λ ≤ δ. A digraph is maximally edge-connected if λ = δ. A digraph is called super-edge-connected if every minimum edge-cut consists of edges incident to or from a vertex of minimum degree. In this note we show that a digraph is maximally edge-connected or super-edge-connected if the number of arcs is large enough.

A connected graph [Formula: see text] is super edge-connected, if every minimum edge-cut of [Formula: see text] is the set of edges incident with a vertex. In this paper, the concept of super edge-connectivity of graphs is generalized to hypergraphs and a necessary and sufficient condition for an [Formula: see text]-uniform and linear hypergraph with diameter at most 2 to be super edge-connected is given.

On the complexity of making a distinguished vertex minimum or maximum degree by vertex deletion

The augmented eccentric connectivity index of a graph which is a generalization of eccentric connectivity index is defined as the summation of the quotients of the product of adjacent vertex degrees and eccentricity of the concerned vertex of a graph. In this paper we established some relationships between augmented eccentric connectivity index and several other graph invariants like number of vertices, number of edges, maximum vertex degree, minimum vertex degree, the total eccentricity index the first Zagreb indices and the second multiplicative Zagreb index.

The atom-bond sum-connectivity (ABS) index of a graph G G with edges e 1 , … , e m {e}_{1},\ldots ,{e}_{m} is the sum of the numbers 1 − 2 ( d e i + 2 ) − 1 \sqrt{1-2{\left({d}_{{e}_{i}}+2)}^{-1}} over 1 ≤ i ≤ m 1\le i\le m , where d e i {d}_{{e}_{i}} is the number of edges adjacent to e i {e}_{i} . In this article, we study the maximum values of the ABS index over graphs with given parameters. More specifically, we determine the maximum ABS index of connected graphs of a given order with a fixed (i) minimum degree, (ii) maximum degree, (iii) chromatic number, (iv) independence number, or (v) number of pendent vertices. We also characterize the graphs attaining the maximum ABS values in all of these classes.

Shortest Fuzzy Hamiltonian Cycle on Transportation Network Using Minimum Vertex Degree and Time-dependent Dijkstra’s Algorithm

International audience

A graph is said to be edge-superconnected if each minimum edge-cut consists of all the edges incident with some vertex of minimum degree. A graph G is said to be a $\{d,d+1\}$ **image** -semiregular graph if all its vertices have degree either d or $d+1$ **image** . A smallest $\{d,d+1\}$ **image** -semiregular graph G with girth g is said to be a $(\{d,d+1\};g)$ **image** -cage. We show that every $(\{d,d+1\};g)$ **image** -cage with odd girth g is edge-superconnected. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011 © 2011 Wiley Periodicals, Inc.

Non-Singular graphs with a Singular Deck

Bounds and extremal graphs for the energy of complex unit gain graphs

The first Zagreb index M1 (G) defined as the sum of squares of the degrees of the vertices. In this paper we compare and analyze numerous lower bounds for the first Zagreb index involving the number of vertices, the number of edges and the maximum and minimum vertex degree. In addition, we propose new lower bound and correct the equality case in [E.I. Milovanović and I.Ž. Milovanović, Sharp Bounds for the first Zagreb index and first Zagreb coindex, Miskolc Mathematical notes, 16 (2015) 1017-1024].

Define the zeroth-order Randic index $$R^0(G)=\sum _{x\in V(G)}\frac{1}{\sqrt{d_G(x)}}$$, where $$d_G(x)$$ denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super-edge-connected in terms of the zeroth-order Randic index, respectively.

Describing the neighborhoods of 5-vertices in 3-polytopes with minimum degree 5 and no vertices of degree from 6 to 8

A lower bound for the smallest uniquely hamiltonian planar graph with minimum degree three