Find in Library

Graph connectivity theory is important in network implementations, transportation, network routing and network tolerance, among other things. Separation edges and vertices refer to single points of failure in a network, and so they are often sought-after. Chandramouleeswaran et al. introduced the principle of semiring valued graphs, also known as <i>S</i>-valued symmetry graphs, in 2015. Since then, works on <i>S</i>-valued symmetry graphs such as vertex dominating set, edge dominating set, regularity, etc. have been done. However, the connectivity of <i>S</i>-valued graphs has not been studied. Motivated by this, in this paper, the concept of connectivity in <i>S</i>-valued graphs has been studied. We have introduced the term vertex <i>S</i>-connectivity and edge <i>S</i>-connectivity and arrived some results for connectivity of a complete <i>S</i>-valued symmetry graph, <i>S</i>-path and <i>S</i>-star. Unlike the graph theory, we have observed that the inequality for connectivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><msup><mi>κ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> holds in the case of <i>S</i>-valued graphs only when there is a symmetry of the graph as seen in Examples 3–5.