TOTAL INDUCED VERTEX STRESS IN BARBELL-LIKE GRAPHS

This paper introduces new parameters called induced vertex stress and total induced vertex stress in G, respectively. For graphs G and H, aspects of the maximum and minimum total induced vertex stress that can be obtained by 1-edge addition and 2-vertex merging are discussed.


Introduction
It is assumed that the reader is familiar with the general notation and concepts in graphs. See [1,2,6]. Only finite, connected and undirected simple graphs will be considered throughout this paper. In 1953, the researcher Alfonso Shimbel [4] introduced the notion of vertex stress in a graph G denoted by S G (v), v ∈ V (G) (see [4]). Recall that the vertex stress of vertex v ∈ V (G) is the number of times v is contained as an internal vertex in all the shortest paths between all pairs of distinct vertices in V (G)\v. Formally are only aware of two papers in which the notion of vertex stress has been furthered. See [3,5].
In [3] the notion of the induced vertex stress by v 1 on each of v i , i = 2, 3, 4, . . . , v n on a path v 1 v 2 v 3 · · · v n was introduced. The induced vertex stress by v 1 is given by, ∫ v1 (v i ) = n − i. Therefore, the total vertex stress induced by v 1 equals, . We now generalize this notion for a graph G.
It naturally follows that the total induced vertex stress of graph G is s( Since we account for the shortest paths between an ordered vertex pair it follows that s(G) is even. Therefore, an even number of vertices v i may exist for which s G (v i ) is an odd value. The aforesaid reminds of the classical result with regards to the number of vertices with odd degree. We also have a useful result which links this notion to the parameter of total vertex stress in the graph G.
Proof. Definition 1.1 implies that for a specific shortest is an ordered pair, it follows that for the specific inverse (v j , v i )-path, the same term is recounted. Hence, throughout the graph G, each value S G (v i ), i = 1, 2, 3, . . . , n is counted twice. Therefore, the result s(G) = 2S(G).
Theorem 1.2 implies that for a stress regular graph [5], We are now ready to move to the main sections of this paper. The study focuses on the addition of an edge between two graphs as well as the merging of two vertices. Though the graph operations seem trivial, they reveal a complicated change in the total induced vertex stress. An important analytical application is found amongst others, in the field of mathematical chemistry. Many compound chemical structures are simply the addition of an edge between pairwise principle molecules (or nuclei) or chained molecules.

Total induced vertex stress of type 1 barbell-like graphs
Recall that a barbell graph is a graph obtained by taking two copies of C n , n ≥ 3 and joining them with an edge. This idea can be generalized by joining any two graphs G and H with an edge. This family of graphs (infinitely many) is called the Type 1 barbell-like graphs. By convention, if u i ∈ V (G) and v j ∈ V (H) then joining the graphs with edge u i v j will be denoted by G H(u i , v j ) or equivalently by H G(v j , u i ).
Theorem 2.1. For graphs G and H of order n and m respectively, and Proof. For the purpose of this proof we first prove the one-way partial result i.e. the total induced vertex stress from V (G) to V (H). We say that vertex u t induces vertex stress on transiting to vertex v s along all the shortest (u t , v s )-paths, In addition we have the values s(G) and s(H) to begin with. Adding all parts and simplifying yield the result.
A more complex graph can now be addressed. Consider the graphs The result from Theorem 2.1 can be applied to recursively determine the total induced vertex stress. However, with each recursive step the vertex stress induced by each vertex must be updated accordingly. The next corollary from the proof of Theorem 2.1, provides these new values.
2.1. Maximum and minimum total induced vertex stress. Note that in the result of Theorem 2.1, the terms s(G) and s(H) are constants whilst all other terms may vary by the choice of u i , v j . This observation suggests that over all such distinct pairwise choices, the parameter s(G H(u i , v j )) will attain a maximum and a minimum value.
Proof. For n = 1, 2, 3, 4 and up to isomorphism, there are only 1, 1, 2 and 6 connected simple graphs respectively. It is easy to verify that the result holds for order 1 ≤ n ≤ 4. Now consider any connected simple graph of order n ≥ 5. Since, any such graph G is a subgraph of K n we may begin with K n and through iterative deletion of an edge at a time, reconstruct G. For K n the result holds since Pair the vertices u i and v i in a one-on-one association. The vertices v j which associate with the closed neighborhood N [u i ] will be denoted by N G [v i ]. By convention ι(v j ) = j. Begin with v 1 and from V (K n )\N [v 1 ] select the smallest ι(v k1 ) and delete edge v 1 v k1 . Clearly, vertices v 1 , v k1 each induce a total of vertex stress equal to n − 2. Hence, for Q 1, If deg Q1,r (v 2 ) > deg G (u 2 ), follow the described procedure to delete edges. When deg Q2,q (v 2 ) = deg G (u 2 ) is reached, then two possibilities exist. Firstly, if deg G (u 1 . However, for both cases the result holds i.e. for v 1 or v 2 since both, deg G (u 1 ) = deg G (u 2 ) = δ(G). Through immediate induction, the result holds for G of order n. Since G and n were random, it follows by imbedded induction that the result i.e. if deg G (u i ) ≤ deg G (u j ) then s G (u i ) ≥ s G (u j ) holds for all graphs G of order n ≥ 1.
For the application of Theorem 2.3, we note specific interpretations. For a graph G of order n ≥ 1 it follows that: Note that (b) is a corollary from Theorem 2.3 which the reader may prove formally. Consider the sets X = {u i : s G (u i ) is a maximum in G} and Y = {u j : s G (u j ) is a minimum in G}.
Lemma 2.4. For a graph G of order n ≥ 1 let k G (i, j) be the number of distinct shortest (u i , u j )-paths, 1 ≤ j =i ≤ n of length G (i, j). It follows that: Proof. Both (a) and (b) follow from the fact that k G (i, j) is well-defined for all u i ∈ V (G) and Theorem 2.3 yields the other required term.
(b) For graphs G and H of order n and m respectively, let u i ∈ V (G) be such that Proof. The result is a direct consequence of Theorem 2.3 and Lemma 2.4.

Total induced vertex stress of type 2 barbell-like graphs
Joining any two graphs G and H by merging vertex u i ∈ V (G) with vertex v j ∈ V (H) results in a Type 2 barbell-like graph. By convention, such new graph is denoted by, G H(u i , v j ) or equivalently by H G(v j , u i ).
Theorem 3.1. For graphs G and H of order n and m respectively, and u i ∈ V (G), v j ∈ V (H) and k G (t, i) the number of distinct shortest (u t , u i )-paths from u t to u i in G and similarly, k H (t, j) the number of distinct shortest (v t , v j )-paths in H it follows that, Proof. Suffice to say that the result follows from the result of Theorem 2.1 by contracting the edge u i v j .
A more complex graph can now be addressed. Consider the graphs H i , 1 ≤ i ≤ and let G = ((((H 1 H 2 (u i , v j )) H 3 (v t , w s )) · · · ) H (y q , z r )). Put differently, for graphs H i , i = 1, 2, 3, . . . , and The result from Theorem 3.1 can be applied to recursively determine the total induced vertex stress. However, with each recursive step the vertex stress induced by each vertex must be updated accordingly. The next corollary from the proof of Theorem 3.1 provides these new values.
(b) For graphs G and H of order n and m respectively, let u i ∈ V (G) be such that

Conclusion
Consider the complete graph K n , n ≥ 2 as well as graphs G j , j = 1, 2, 3, . . . , n. Merge any vertex u i ∈ V (G j ) with vertex v j ∈ K n . Hence, K n is a cut-clique in the compound graph H 1 we constructed. If all edges of K n are contracted a graph H 2 is obtained which contains a compound cut vertex. The study of vertex stress (atom stress) and total vertex stress in a graph (molecular stress) and the notion of induced vertex stress (induced atomic stress) in chemical structures renders wide scope for further research. It is hoped that such research will contribute to a deeper understanding of chemical properties and contribute to the field of mathematical chemistry.