A path <i>π</i> = [<i>v</i><sub>1</sub>, <i>v</i><sub>2</sub>, …, <i>v</i><sub><em>k</em></sub>] in a graph <i>G&...A path <i>π</i> = [<i>v</i><sub>1</sub>, <i>v</i><sub>2</sub>, …, <i>v</i><sub><em>k</em></sub>] in a graph <i>G</i> = (<i>V</i>, <i>E</i>) is an uphill path if <i>deg</i>(<i>v</i><sub><i>i</i></sub>) ≤ <i>deg</i>(<i>v</i><sub><i>i</i>+1</sub>) for every 1 ≤ <i>i</i> ≤ <i>k</i>. A subset <i>S </i><span style="white-space:nowrap;"><span style="white-space:nowrap;">⊆</span></span> <i>V</i>(<i>G</i>) is an uphill dominating set if every vertex <i>v</i><sub><i>i</i></sub> <span style="white-space:nowrap;"><span style="white-space:nowrap;">∈</span> </span><i>V</i>(<i>G</i>) lies on an uphill path originating from some vertex in <i>S</i>. The uphill domination number of <i>G</i> is denoted by <i><span style="white-space:nowrap;"><i><span style="white-space:nowrap;"><i>γ</i></span></i></span></i><sub><i>up</i></sub>(<i>G</i>) and is the minimum cardinality of the uphill dominating set of <i>G</i>. In this paper, we introduce the uphill domination polynomial of a graph <i>G</i>. The uphill domination polynomial of a graph <i>G</i> of <i>n</i> vertices is the polynomial <img src="Edit_75fb5c37-6ef5-4292-9d3a-4b63343c48ce.bmp" alt="" />, where <em>up</em>(<i>G</i>, <i>i</i>) is the number of uphill dominating sets of size <i>i</i> in <i>G</i>, and <i><span style="white-space:nowrap;"><i><span style="white-space:nowrap;"><i>γ</i></span></i></span></i><i><sub>up</sub></i>(<i>G</i>) is the uphill domination number of <i>G</i>, we compute the uphill domination polynomial and its roots for some families of standard graphs. Also, <i>UP</i>(<i>G</i>, <em>x</em>) for some graph operations is obtained.展开更多
A graphical index is a numeric value corresponding to a graph which is structurally invariant and in molecular graph theory these invariants are known as topological indices. In the field of Chemical and Medical Scien...A graphical index is a numeric value corresponding to a graph which is structurally invariant and in molecular graph theory these invariants are known as topological indices. In the field of Chemical and Medical Sciences, the topological indices are used to study the chemical, biological, medical and pharmaceutical features of drugs. With reference to the previous deadly diseases, the COVID-19 pandemic has considered to be the biggest life threatening issue that modern medicines have ever tackled. COVID-19 is immedicable and even the existing treatments are only helping the certain group of sufferers. Scientists have tested available antiviral agents and got a favorable impact on recovering from pandemic. Some of these antiviral agents are remdesivir, chloroquine, hydroxychloroquine, theaflavin and dexamethasone. Keeping in view of the importance of topological indices in the study of pharmaceutical and chemical drugs, in this paper, we calculate the <em>M<sub>dn</sub></em>-Polynomial, some downhill Zagreb topological indices and some downhill Zagreb polynomials of some of the anti viral agents remdesivir, chloroquine, hydroxychloroquine, theaflavin and dexamethasone. The results thus obtained may be useful for the finding new medicine and vaccine for the treatment of COVID-19.展开更多
文摘A path <i>π</i> = [<i>v</i><sub>1</sub>, <i>v</i><sub>2</sub>, …, <i>v</i><sub><em>k</em></sub>] in a graph <i>G</i> = (<i>V</i>, <i>E</i>) is an uphill path if <i>deg</i>(<i>v</i><sub><i>i</i></sub>) ≤ <i>deg</i>(<i>v</i><sub><i>i</i>+1</sub>) for every 1 ≤ <i>i</i> ≤ <i>k</i>. A subset <i>S </i><span style="white-space:nowrap;"><span style="white-space:nowrap;">⊆</span></span> <i>V</i>(<i>G</i>) is an uphill dominating set if every vertex <i>v</i><sub><i>i</i></sub> <span style="white-space:nowrap;"><span style="white-space:nowrap;">∈</span> </span><i>V</i>(<i>G</i>) lies on an uphill path originating from some vertex in <i>S</i>. The uphill domination number of <i>G</i> is denoted by <i><span style="white-space:nowrap;"><i><span style="white-space:nowrap;"><i>γ</i></span></i></span></i><sub><i>up</i></sub>(<i>G</i>) and is the minimum cardinality of the uphill dominating set of <i>G</i>. In this paper, we introduce the uphill domination polynomial of a graph <i>G</i>. The uphill domination polynomial of a graph <i>G</i> of <i>n</i> vertices is the polynomial <img src="Edit_75fb5c37-6ef5-4292-9d3a-4b63343c48ce.bmp" alt="" />, where <em>up</em>(<i>G</i>, <i>i</i>) is the number of uphill dominating sets of size <i>i</i> in <i>G</i>, and <i><span style="white-space:nowrap;"><i><span style="white-space:nowrap;"><i>γ</i></span></i></span></i><i><sub>up</sub></i>(<i>G</i>) is the uphill domination number of <i>G</i>, we compute the uphill domination polynomial and its roots for some families of standard graphs. Also, <i>UP</i>(<i>G</i>, <em>x</em>) for some graph operations is obtained.
文摘A graphical index is a numeric value corresponding to a graph which is structurally invariant and in molecular graph theory these invariants are known as topological indices. In the field of Chemical and Medical Sciences, the topological indices are used to study the chemical, biological, medical and pharmaceutical features of drugs. With reference to the previous deadly diseases, the COVID-19 pandemic has considered to be the biggest life threatening issue that modern medicines have ever tackled. COVID-19 is immedicable and even the existing treatments are only helping the certain group of sufferers. Scientists have tested available antiviral agents and got a favorable impact on recovering from pandemic. Some of these antiviral agents are remdesivir, chloroquine, hydroxychloroquine, theaflavin and dexamethasone. Keeping in view of the importance of topological indices in the study of pharmaceutical and chemical drugs, in this paper, we calculate the <em>M<sub>dn</sub></em>-Polynomial, some downhill Zagreb topological indices and some downhill Zagreb polynomials of some of the anti viral agents remdesivir, chloroquine, hydroxychloroquine, theaflavin and dexamethasone. The results thus obtained may be useful for the finding new medicine and vaccine for the treatment of COVID-19.
