# Two classes of graphs related to extremal eccentricities

The reformulated Zagreb indices of a graph are obtained from the original Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of its two end vertices minus 2. In this paper, we obtain two upper bounds of the first reformulated Zagreb index among all graphs with p pendant vertices and all graphs having key vertices for which they will become trees after deleting their one key vertex.

Moreover, the corresponding extremal graphs which attained these bounds are characterized.

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Some constants are used to characterize some properties of the graph of a molecule, which are usually called topological indices. For a graph Gthe first Zagreb index and the second Zagreb index [ 21 ] are defined as. For an edgethe edge degree of e is referred as the sum of degrees of its two end vertices minus 2 and is denoted by.

For a given Glet be its line graph. Observe that two edges are adjacent in G if and only if the corresponding two vertices are adjacent in. The reformulated Zagreb indices, particularly its bounds, have attracted recently the attention of many mathematicians see, [ 1222 — 30 ].

In order to describe this more clearly in the sequel, we now introduce some notations. Let be the set of connected graphs with pendant vertices.

### Extremal graph theory

Evidently, ifthen there will be a connected subgraph with order for which G can be reconstructed by linking p vertices to some vertices. For convenience, we call as the core of G. Since is connected, it has two extremal cases, i. Let be the graph with coreand let all pendants of have a common neighbor in. Let be the set of all graphs for which each of its element will be changed to a tree by deleting some of its vertex. That is to say, if G belongs tothen there is a vertex such that is isomorphic to a tree.

We call the vertex as the key of G. Note that, for a given graph, its key may not be unique, e. Let be the graph with two vertices having degree and other vertices owning degree 2. Obviously, and the two vertices possessing degree are keys. In this paper, we determined the two upper bounds of reformulated Zagreb indices of two kinds of graphs and characterized completely extremal graphs.

In the section, we will research the maximal properties regarding the reformulated Zagreb index on andrespectively. Meanwhile, the graphs attaining the bounds are obtained.

Based on the definition ofthe following result holds obviously. Proposition 1.Extremal graph theory is a branch of mathematics that studies how global properties of a graph influence local substructure.

One of the main objects of study in this area of graph theory are extremal graphswhich are maximal or minimal with respect to some global parameter, and such that they contain or do not contain a local substructure- such as a clique, or an edge coloring.

Question 1. Question 2. Question 3. Question 4. For many particular classes of bipartite graphs, determining the asymptotic behavior remains an open problem. Several foundational results in extremal graph theory answer questions which follow this general formulation:.

Question 5. The extremal examples are trees. The theorems mentioned above can be rephrased in terms of edge density. The theorems stated above give conditions for a small object to appear within a perhaps very large graph.

Another direction in extremal graph theory is looking for conditions that guarantee the existence of a structure that covers every vertex. Edge counting conditions give no indication as to how the edges in the graph are distributed, leading to results which only find bounded structures on very large graphs. This provides motivation for considering the minimum degree parameter, which is defined as.

A large minimum degree eliminates the possibility of having some 'pathological' vertices; if the minimum degree of a graph G is 1, for example, then there can be no isolated vertices even though G may have very few edges. Even though many important observations have been made in the field of extremal graph theory, several questions still remain unanswered. Another important conjecture in extremal graph theory was formulated by Sidorenko in From Wikipedia, the free encyclopedia.

Extremal graph theory, in its strictest sense, is a branch of graph theory developed and loved by Hungarians. Categories : Extremal graph theory. Namespaces Article Talk.

Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file.Graphs come with various properties which are used for characterization of graphs depending on their structures.

These properties are defined in specific terms pertaining to the domain of graph theory. In this chapter, we will discuss a few basic properties that are common in all graphs.

It is number of edges in a shortest path between Vertex U and Vertex V. If there are multiple paths connecting two vertices, then the shortest path is considered as the distance between the two vertices.

There can be any number of paths present from one vertex to other. Among those, you need to choose only the shortest one.

The maximum distance between a vertex to all other vertices is considered as the eccentricity of vertex. The distance from a particular vertex to all other vertices in the graph is taken and among those distances, the eccentricity is the highest of distances. The minimum eccentricity from all the vertices is considered as the radius of the Graph G. The minimum among all the maximum distances between a vertex to all other vertices is considered as the radius of the Graph G.

From all the eccentricities of the vertices in a graph, the radius of the connected graph is the minimum of all those eccentricities. The maximum eccentricity from all the vertices is considered as the diameter of the Graph G. The maximum among all the distances between a vertex to all other vertices is considered as the diameter of the Graph G.

### Graph Theory - Basic Properties

If the eccentricity of a graph is equal to its radius, then it is known as the central point of the graph. In the example graph, the circumference is 6, which we derived from the longest cycle a-c-f-g-e-b-a or a-c-f-d-e-b-a. Graph Theory - Basic Properties Advertisements. Previous Page. Next Page. Previous Page Print Page. Dashboard Logout.Steve, I love the post. I troll for good tips like these all the time but had not seen any of the ones you noted in your post so it is like finding the end of the rainbow.

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You can limit the salt in your meals and still have food that tastes great. Start with these four tips:Packaged foods often have a lot of salt in them, since salt is a preservative. Cooking your own meals is the best way to control how much sodium goes into your food.

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## Note on the Reformulated Zagreb Indices of Two Classes of Graphs

For side dishes, make brown rice or whole baked sweet or white potatoes instead of instant or flavored rice or mashed potatoes. Check every food label for sodium content. Look for products labeled "sodium-free" (less than 5 milligrams of sodium per serving), "very low sodium" (35 milligrams or less per serving), or "no salt added.

Watch for additives such as monosodium glutamate (MSG), baking soda, baking powder, and sodium benzoate, which are all sodium compounds. When a recipe calls for even a pinch of salt, replace it with another herb or spice. You can add rosemary, marjoram, thyme, tarragon, onion powder, garlic powder, pepper, nutmeg, cumin, curry powder, ginger, cilantro, bay leaf, oregano, dry mustard, or dill.

Or sprinkle in a commercial salt-free seasoning blend. You can use dozens of different ingredients to add flavor to your low-sodium meals. Here are just a few ideas for common foods:U. Department of Health and Human Services and U. Department of Agriculture: "Dietary Guidelines for Americans - 2005," "Dietary Guidelines for Americans, 2010. Department of Health and Human Services: "Your Guide to Lowering Your Blood Pressure with DASH.

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Gene Discovery May Help Fight Alzheimer's FDA Warns Biotin Can Interfere With Lab Tests Is It Time to Discontinue the Pap Test. Start with these four tips: 1. Cook It Yourself Packaged foods often have a lot of salt in them, since salt is a preservative. Check on Salty Items Check every food label for sodium content. Don't Follow Directions When a recipe calls for even a pinch of salt, replace it with another herb or spice.

Here are just a few ideas for common foods: Marinate chicken breasts or pork chops in lemon juice, orange juice, or wine. Roll fish in sesame seeds before baking. Spice up beef with a mixture of onion, peppers, sage, and thyme.

Simmer carrots in cinnamon and nutmeg. Sprinkle some dill and parsley onto potatoes before roasting. Add a dash of chili powder to corn. Toss your pasta with fresh chopped garlic. WebMD Medical Reference Reviewed by Kathleen M.

Zelman, MPH, RD, LD on April 28, 2016 SOURCES: U. BMJ, published online April 20, 2007.

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