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Alzheimer’s Disease (AD) is an incurable neurodegenerative disorder which mainly affects older adults. An early diagnosis is essential because medical treatments can slow down the progression of the disease only if provided during the first stage, called Mild Cognitive Impairment (MCI). Starting from the study of electroencephalografic signals, brain functional connectivity analyses can be performed with the support of the graph theory. In particular, the purpose of this work is to verify the performances of three indexes, typically adopted to evaluate the graph robustness, in order to estimate the functional connectivity for three groups of subjects: healthy controls and people affected by dementia at two different stages (MCI and AD). The results obtained by the Connection Density Index, the Randić Index, and a normalized version of the Kirchhoff Index revealed a higher robustness in the brain networks of healthy people, followed by MCI and, finally, by AD patients, consistent with the hallmarks of Alzheimer’s disease. The statistical analysis showed that there is a significant difference between controls and AD for all three indexes. Finally, all three indexes were compared, revealing that the the Randić Index outperformed the other two indexes. These preliminary outcomes will be exploited to address further in-depth and time-expensive analyses for improving the diagnosis of Alzheimer’s disease.

Alzheimer’s Disease (AD) is an incurable neurodegenerative disorder which mainly affects older adults. An early diagnosis is essential because medical treatments can slow down the progression of the disease only if provided during the first stage, called Mild Cognitive Impairment (MCI). Starting from the study of electroencephalografic signals, brain functional connectivity analyses can be performed with the support of the graph theory. In particular, the purpose of this work is to verify the performances of three indexes, typically adopted to evaluate the graph robustness, in order to estimate the functional connectivity for three groups of subjects: healthy controls and people affected by dementia at two different stages (MCI and AD). The results obtained by the Connection Density Index, the Randić Index, and a normalized version of the Kirchhoff Index revealed a higher robustness in the brain networks of healthy people, followed by MCI and, finally, by AD patients, consistent with the hallmarks of Alzheimer’s disease. The statistical analysis showed that there is a significant difference between controls and AD for all three indexes. Finally, all three indexes were compared, revealing that the the Randić Index outperformed the other two indexes. These preliminary outcomes will be exploited to address further in-depth and time-expensive analyses for improving the diagnosis of Alzheimer’s disease.

© 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0, For a given graph G = (V, E), the degree mean rate of an edge uv ¿ E is a half of the quotient between the geometric and arithmetic means of its end-vertex degrees d(u) and d(v). In this note, we derive tight bounds for the Randic index of G in terms of its maximum and minimum degree mean rates over its edges. As a consequence, we prove the known conjecture that the average distance is bounded above by the Randic index for graphs with order n large enough, when the minimum degree d is greater than (approximately) ¿1/3 , where ¿ is the maximum degree. As a by-product, this proves that almost all random (Erdos–Rényi) graphs satisfy the conjecture, Peer Reviewed, Postprint (author's final draft)

For a connected undirected graph $G=(V,E)$ with vertex set $\{1, 2, \ldots, n\}$ and degrees $ d_i$, for $1\le i \le n$, we show that $$ABC(G) \le \sqrt{(n-1)(|E|-R_{-1}(G))},$$ where $\displaystyle R_{-1}(G)=\sum_{(i,j)\in E}\frac{1}{d_id_j}$ is the Randi\'c index. This bound allows us to obtain some maximal results for the $ABC$ index with elementary proofs and to improve all the upper bounds in [20], as well as some in [17], using lower bounds for $R_{-1}(G)$ found in the literature and some new ones found through the application of majorization.