Graph Theory Parameters Research Articles

Zero Forcing Domination in Some Graphs: Characterizations and Derived Formulas

In this paper, we initiate the study on zero forcing domination in a graph and we characterize the zero forcing dominating sets in some special graphs and the join of two graphs. We derive some formulas of the parameter using characterization results. Moreover, we present some relationships of this parameter with other known parameters in graph theory.

Characteristics of Functional Connections and Topographical Properties Distinguished the Healthy State and Obsessive-Compulsive Disorder

Background: Obsessive-compulsive disorder (OCD) is characterized by alterations in brain connectivity, particularly within the default mode network (DMN) and the salience network (SN). Investigating these connectivity differences can provide a deeper understanding of the neural mechanisms underlying OCD. Methods: This cross-sectional study involved 58 patients diagnosed with OCD and 38 healthy control subjects, totaling 96 participants. Resting-state functional MRI (fMRI) data were acquired and analyzed using the CONN toolbox to examine functional connectivity within intrinsic resting-state networks. Graph theory metrics were applied to evaluate node connections and the overall network topology. Clinical symptoms were assessed using the Yale-Brown Obsessive-Compulsive Scale (Y-BOCS), and their correlations with connectivity patterns and graph-theory parameters were analyzed. Results: The OCD patients and healthy controls were matched in terms of age, gender, marital status, socioeconomic status, and handedness. However, OCD patients had significantly worse general health, quality of life, and higher levels of depression and anxiety. Network analyses revealed altered whole-brain connectivity in OCD patients, particularly within the DMN and the frontoparietal network. The most significant between-group connectivity differences were observed between the posterior parietal cortex (PPC) and the precuneus. Disruptions in the DMN, specifically altered connectivity between the medial prefrontal cortex and posterior cingulate cortex, and changes in the SN involving the anterior insula and anterior cingulate cortex, were significantly correlated with the severity of OCD symptoms. Conclusions: The findings suggest that OCD is associated with distinct alterations in DMN connectivity, which may play a critical role in the disorder's pathophysiology. These disruptions offer potential targets for therapeutic intervention. Further research is needed to explore these connectivity changes in larger cohorts and at various stages of OCD to better understand their clinical significance.

P11.05.B LONGITUDINAL MODIFICATIONS OF GRADIENTS AND GRAPH-THEORY MEASURES OF FUNCTIONAL MRI CONNECTIVITY IN GLIOMA PATIENTS AS POTENTIAL MARKERS OF POST-SURGICAL LONG-TERM COGNITIVE OUTCOME

Abstract BACKGROUND Integrating connectomics and functional brain activity into the study of brain tumors is among neuro-oncology’s challenges. Functional MRI (fMRI) studies showed that gliomas are integrated with brain networks’ activity, exerting effects beyond their margins. We investigated longitudinal modifications of fMRI gradients and graph-theory parameters in glioma patients, assessing their role in cognitive prognostication. MATERIAL AND METHODS Diffuse non-enhancing glioma patients underwent fMRI and cognitive assessments (OCS-Bridge) before, within 72h after, 3 and 12 months after surgery. Cognitive data were normalized and averaged into a global score. Differences between the last and preoperative scores were computed (“cognitive difference”). fMRI data underwent preprocessing, cortical parcellation, computation of timeseries, connectivity matrices, gradients and graph-theory measures. To quantify gradient dispersion, we computed a circularity metric (“CM”) of the hull perimeter of the first two gradients’ distribution. T-tests, partial correlations (correcting for volume and surgery radicality) and linear regression (regressors: age, volume, preoperative cognition, surgery radicality) were used to investigate associations with cognitive difference. RESULTS 17 patients (22-56 years, 8 females) were included, 4 had long-term cognitive worsening. Postoperative circularity metric, especially of the healthy intra- and inter-hemispheric connectivity (“HIIHC”), was higher than the preoperative (0.43 vs 0.32, t=2.4, p=0.03). Month3-postoperative and month3-preoperative differences in HIIHC CM were higher in patients with compared to those without cognitive worsening (0.33 vs -0.06, t=6.9, p<0.0001 and 0.27 vs 0.009, t=4.3, p=0.001). Both were negative predictors of the cognitive difference (the first: R=-0.94 with p=0.0002; β=-1.1 with p=0.001 and regression model’s p=0.003; the second: R=-0.76 with p=0.006; β=-1.1 with p=0.02 and model’s p=0.05). Month3-preoperative network modularity difference correlated with the second metric (R=-0.72, p=0.01) and the cognitive difference (R=0.67, p=0.02), while the preoperative modularity with the first metric (R=0.77, p=0.014). Similar results were obtained using a predetermined number of modules (Yeo’s 7 networks). Month12-preoperative difference in healthy intra-hemispheric connectivity CM also correlated with cognitive difference (R=-0.91, p=0.002). CONCLUSION Functional gradients seem dispersed after glioma surgery. Their recovery or worsening at month 3 could predict opposite long-term cognitive outcomes, putatively due to connectivity changes mainly involving the healthy hemisphere. Low preoperative network modularity and its increase at month 3 are associated with such gradient dispersion. Cognitive rehabilitation might be beneficial in the early post-surgical phase.

Some Properties of Zero Forcing Hop Dominating Sets in a Graph

In this paper, we initiate the study of a zero forcing hop domination in a graph. We establish some properties of this parameter and we determine its connections with other known parameters in graph theory. Moreover, we obtain some exact values or bounds of the parameter on the generalized graph, some families of graphs, and graphs under some operations via characterizations.

Vertex Cover Hop Dominating Sets in Graphs

Let $G$ be a graph. Then a subset $C$ of vertices of $G$ is called a vertex cover hop dominating if $C$ is both a vertex cover and a hop dominating of $G$. The vertex cover hop domination number of $G$, denoted by $\gamma_{vch}(G)$, is the minimum cardinality among all vertex cover hop dominating sets in $G$. In this paper, we initiate the study of vertex cover hop domination in a graph and we determine its relations with other parameters in graph theory. We characterize the vertex cover hop dominating sets in some special graphs, join, and corona of two graphs and we finally obtain the exact values or bounds of the parameters of these graphs.

Metric based resolvability of cycle related graphs

<abstract><p>If a subset of vertices of a graph, designed in such a way that the remaining vertices have unique identification (usually called representations) with respect to the selected subset, then this subset is named as a metric basis (or resolving set). The minimum count of the elements of this subset is called as metric dimension. This concept opens the gate for different new parameters, like fault-tolerant metric dimension, in which the failure of any member of the designed subset is tolerated and the remaining subset fulfills the requirements of the resolving set. In the pattern of the resolving sets, a concept was introduced where the representations of edges must be unique instead of vertices. This concept was called the edge metric dimension, and this as well as the previously mentioned concepts belong to the idea of resolvability parameters in graph theory. In this paper, we find all the above resolving parametric sets of a convex polytope $ {F}_{♃} $ and compare their cardinalities.</p></abstract>

Functional network properties in schizophrenia and bipolar disorder assessed with high-density electroencephalography

BackgroundThe study of the cortical functional network properties in schizophrenia (SZ) may benefit from the use of graph theory parameters applied to high-density electroencephalography (EEG). Connectivity Strength (CS) assesses global synchrony of the network, and Shannon Graph Complexity (SGC) summarizes the network distribution of link weights and allows distinguishing between primary and secondary pathways. Their joint use may help in understanding the underpinnings of the functional network hyperactivation and task-related hypomodulation previously described in psychoses. MethodsWe used 64-sensor EEG recordings during a P300 oddball task in 128 SZ patients (96 chronic, CR, and 32 first episodes, FE), as well as 46 bipolar disorder (BD) patients, and 92 healthy controls (HC). Pre-stimulus and modulation (task-response minus pre-stimulus windows values) of CS and SGC were assessed in the theta band (4–8 Hz) and the broadband (4–70 Hz). ResultsCompared to HC, SZ patients (CR and FE) showed significantly higher pre-stimulus CS values in the broadband, and both SZ and BD patients showed lower theta-band CS modulation. SGC modulation values, both theta-band and broadband, were also abnormally reduced in CR patients. Statistically significant relationships were found in the theta band between SGC modulation and both CS pre-stimulus and modulation values in patients. CS altered measures in patients were additionally related to their cognitive outcome and negative symptoms. A primary role of antipsychotics in these results was ruled out. ConclusionsOur results linking SGC and CS alterations in psychotic patients supported a hyperactive and hypomodulatory network mainly involving connections in secondary pathways.

Characterizations of $J$-Total Dominating Sets in Some Special Graphs and Graphs under Some Operations

Let G be a graph with no isolated vertex. A subset M ⊆ V (G) is called a J-open set if NG(a)\NG(b) ̸= ∅ and NG(b)\NG(a) ̸= ∅ ∀ a, b ∈ M, where a ̸= b. If in addition, M is a total dominating in G, then we call M a J-total dominating set in G. The maximum cardinality amongall J-total dominating set in G, denoted by γJt(G), is called the J-total domination number of G. In this paper, we characterize J-total dominating sets in some special graphs and join of two graphs, and we use these results to obtain formulas for the parameters of these graphs. Moreover, we determine its relationships with other known parameters in graph theory. Finally, we derive the lower bound of the parameter for the corona of two graphs.

$J^2$-Hop Domination in Graphs: Properties and Connections with Other Parameters

A subset $T=\{v_1, v_2, \cdots, v_m\}$ of vertices of an undirected graph $G$ is called a $J^2$-set if$N_G^2[v_i]\setminus N_G^2[v_j]\neq \varnothing$ for every $i\neq j$, where $i,j\in\{1, 2, \ldots, m\}$.A $J^2$-set is called a $J^2$-hop dominating in $G$ if for every $a\in V(G)\s T$, there exists $b\in T$ such that $d_G(a,b)=2$. The $J^2$-hop domination number of $G$, denoted by $\gamma_{J^2h}(G)$, is the maximum cardinality among all $J^2$-hop dominating sets in $G$. In this paper, we introduce this new parameter and wedetermine its connections with other known parameters in graph theory. We derive its bounds with respect to the order of a graph and other known parameters on a generalized graph, join and corona of two graphs. Moreover,we obtain exact values of the parameter for some special graphs and shadow graphs using the characterization results that are formulated in this study.

Graph-based analysis of EEG for schizotypy classification applying flicker Ganzfeld stimulation

Ganzfeld conditions induce alterations in brain function and pseudo-hallucinatory experiences, particularly in people with high positive schizotypy. The current study uses graph-based parameters to investigate and classify brain networks under Ganzfeld conditions as a function of positive schizotypy. Participants from the general population (14 high schizotypy (HS), 29 low schizotypy (LS)) had an electroencephalography assessment during Ganzfeld conditions, with varying visual activation (8 frequencies of random light flicker) and soundscape-induced mood (neutral, serenity, and anxiety). Weighted functional networks were computed in six frequency sub-bands (delta, theta, alpha-low, alpha-high, beta, and gamma) as a function of light-flicker frequency and mood. The brain network was analyzed using graph theory parameters, including clustering coefficient (CC), strength, and global efficiency (GE). It was found that the LS groups had higher CC and strength than the HS groups, especially in bilateral temporal and frontotemporal brain regions. Moreover, some decreases in CC and strength measures were found in LS groups among occipital and parieto-occipital brain regions. LS groups also had significantly higher GE in all Ganzfeld conditions compared to the HS groups. The random under-sampling boosting (RUSBoost) algorithm achieved the best classification performance with an accuracy of 95.34%, specificity of 96.55%, and sensitivity of 92.85% during an anxiety-induction Ganzfeld condition. This is the first exploration of the relationship between brain functional state changes under Ganzfeld conditions in individuals who vary in positive schizotypy. The accuracy of graph-based parameters in classifying brain states as a function of schizotypy is shown, particularly for brain activity during anxiety induction, and should be investigated in psychosis.

On graphs that contain exactly k copies of a subgraph, and a related problem in search theory

We study exak(n,F), the largest number of edges in an n-vertex graph that contains exactly k copies of a given subgraph F. The case k=0 is the Turán number ex(n,F) that is among the most studied parameters in extremal graph theory. We show that for any F and k, exak(n,F)=(1+o(1))ex(n,F) and determine the exact values of exak(n,K3) and exa1(n,Kr) for n large enough. We also explore a connection to the following well-known problem in search theory. We are given a graph of order n that consists of an unknown copy of F and some isolated vertices. We can ask pairs of vertices as queries, and the answer tells us whether there is an edge between those vertices. Our goal is to describe the graph using as few queries as possible. Aigner and Triesch in 1990 showed that the number of queries needed is at least n2−exa1(n,F). Among other results we show that the number of queries that were answered NO is at least n2−exa1(n,F).

More extensive structural damage in temporal lobe epilepsy with hippocampal sclerosis type 1

ObjectiveTo explore clinical and structural differences between mesial temporal lobe epilepsy (mTLE) patients with different hippocampal sclerosis (HS) subtypes. MethodsHigh-resolution T1-weighted MRI and diffusion tensor imaging data were obtained in 41 refractory mTLE patients and 52 age- and sex-matched healthy controls. Postoperative histopathological examination confirmed HS type 1 in 30 patients and HS type 2 in eleven patients. Clinical features, postoperative seizure outcomes, hippocampal subfields volumes, fractional anisotropy (FA) values of white matter regions and graph theory parameters were explored and compared between the HS type 1 and HS type 2 groups. ResultsNo significant differences in clinical features and postsurgical seizure outcomes were found between the HS type 1 and type 2 groups. However, the HS type 1 group showed extra atrophy in ipsilateral parasubiculum than healthy controls and more severe atrophy in contralateral hippocampal fissure than the HS type 2 group. More extensive FA decrease were also observed in the HS type 1 group, involving ipsilateral optic radiation, superior fronto-occipital fasciculus, contralateral uncinate fasciculus, tapetum, bilateral hippocampal cingulum, corona radiata, etc. Furthermore, in spite of similar impairments in characteristic path length, global efficiency and local efficiency in two HS groups, the HS type 1 group showed additional decrease of clustering coefficient than healthy controls. ConclusionsHS type 1 and 2 groups had similar clinical characteristics and postoperative seizure outcomes. More widespread neuronal cell loss in the HS type 1 group contributed to more extensive structural damage and connectivity abnormality. These results shed new light on the imaging correlates of different HS pathology.

Some Properties and Realization Problems Involving Connected Outer-hop Independent Hop Domination in Graphs

In this paper, we construct a realization problems involving connected outer-hop independent hop domination and we determine its connections with other known parameters in graph theory. In particular, given two positive integers a and b with 2 ≤ a ≤ b are realizable as the connected hop domination, connected outer-hop independent hop domination, and connected outer-independent hop domination numbers, respectively, of a connected graph. In addition, we characterize the connected outer-hop independent hop dominating sets in some families of graphs, join and corona of two graphs, and we use these results to derive formulas for the parameters of these graphs.

Identifying properties of pattern completion neurons in a computational model of the visual cortex.

Neural ensembles are found throughout the brain and are believed to underlie diverse cognitive functions including memory and perception. Methods to activate ensembles precisely, reliably, and quickly are needed to further study the ensembles' role in cognitive processes. Previous work has found that ensembles in layer 2/3 of the visual cortex (V1) exhibited pattern completion properties: ensembles containing tens of neurons were activated by stimulation of just two neurons. However, methods that identify pattern completion neurons are underdeveloped. In this study, we optimized the selection of pattern completion neurons in simulated ensembles. We developed a computational model that replicated the connectivity patterns and electrophysiological properties of layer 2/3 of mouse V1. We identified ensembles of excitatory model neurons using K-means clustering. We then stimulated pairs of neurons in identified ensembles while tracking the activity of the entire ensemble. Our analysis of ensemble activity quantified a neuron pair's power to activate an ensemble using a novel metric called pattern completion capability (PCC) based on the mean pre-stimulation voltage across the ensemble. We found that PCC was directly correlated with multiple graph theory parameters, such as degree and closeness centrality. To improve selection of pattern completion neurons in vivo, we computed a novel latency metric that was correlated with PCC and could potentially be estimated from modern physiological recordings. Lastly, we found that stimulation of five neurons could reliably activate ensembles. These findings can help researchers identify pattern completion neurons to stimulate in vivo during behavioral studies to control ensemble activation.

Some upper and lower bounds for $D_{\alpha}$-energy of graphs

The generalized distance matrix of a connected graph $G$, denoted by $D_{\alpha}(G)$, is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G), ~~~~ 0\leq \alpha\leq 1$. Here, $D(G)$ is the distance matrix and $Tr(G)$ represents the vertex transmissions. Let $\partial_{1}\geq \partial_{2}\geq \cdots \geq \partial_{n}$ be the eigenvalues of $D_{\alpha}(G)$ and let $W(G)$ be the Wiener index. The generalizeddistance energy of $G$ can be defined as $E^{D_{\alpha}}(G)=\displaystyle\sum_{i=1}^{n}\left|\partial_i-\frac{2\alpha W(G)}{n}\right|$. In this paper, we develop some new theory regarding the generalized distance energy $E^{D_{\alpha}}(G)$ for a connected graph $G$. We obtain some sharp upper and lower bounds for$E^{D_{\alpha}}(G)$ connecting a wide range of parameters in graph theory including the maximum degree $\Delta$, the Wiener index $W(G)$, the diameter $d$, the transmission degrees, and the generalized distance spectral spread $D_{\alpha}S(G)$. We characterized the special graph classes that attain the bounds.

Clique Centrality and Global Clique Centrality of Graphs

We formally introduce in this paper two parameters in graph theory, namely, clique centrality and global clique centrality. Let G be a finite, simple and undirected graph of order n. A clique in G is a nonempty subset W \(\subseteq\) V (G) such that the subgraph \(\langle\)W\(\rangle\)G induced by W is complete. The maximum size of any clique containing vertex u \(\in\) V (G) is called the clique centrality of u in G. Normalizing the sum of the clique centralities of all the vertices of G will lead us to the global clique centrality of G, whose value ranges from \(\frac{1}{m}\) to 1. In this paper, we study some general properties of the global clique centrality and then evaluate it for some parameterized families of graphs.

The combination of hyperventilation test and graph theory parameters to characterize EEG changes in mild cognitive impairment (MCI) condition.

Hyperventilation (HV) is a voluntary activity that causes changes in the neuronal firing characteristics noticeable in the electroencephalogram (EEG) signals. HV-related changes have been scribed to modulation of pO2/pCO2 blood contents. Therefore, an HV test is routinely used for highlighting brain abnormalities including those depending to neurobiological mechanisms at the basis of neurodegenerative disorders. The main aim of the present paper is to study the effectiveness of HV test in modifying the functional connectivity from the EEG signals that can be typical of a prodromal state of Alzheimer's disease (AD), the Mild Cognitive Impairment prodromal to Alzheimer condition. MCI subjects and a group of age-matched healthy elderly (Ctrl) were enrolled and subjected to EEG recording during HV, eyes-closed (EC), and eyes-open (EO) conditions. Since the cognitive decline in MCI seems to be a progressive disconnection syndrome, the approach we used in the present study is the graph theory, which allows to describe brain networks with a series of different parameters. Small world (SW), modularity (M), and global efficiency (GE) indexes were computed among the EC, EO, and HV conditions comparing the MCI group to the Ctrl one. All the three graph parameters, computed in the typical EEG frequency bands, showed significant changes among the three conditions, and more interestingly, a significant difference in the GE values between the MCI group and the Ctrl one was obtained, suggesting that the combination of HV test and graph theory parameters should be a powerful tool for the detection of possible cerebral dysfunction and alteration.

Limbic covariance network alterations in patients with transient global amnesia.

We compared limbic structure volumes and graph theory parameters of the limbic covariance network between patients with transient global amnesia (TGA) and healthy controls, and between patients with single and recurrent TGA events. We retrospectively enrolled 122 patients with TGA (single event, n = 107; recurrent events, n = 15) and 50 healthy controls who underwent three-dimensional T1-weighted MRI imaging of the brain. Volumetric analysis of the subcortical limbic structures, including the hippocampus, amygdala, thalamus, mammillary body, hypothalamus, basal forebrain, septal nuclei, fornix, and nucleus accumbens, was performed. We examined the limbic covariance network using a graph theory. Limbic structure volumes did not differ between patients with TGA and healthy controls, and between patients with a single event and those with recurrent events. However, the radius of the limbic covariance network was significantly greater in patients with TGA than in healthy controls (6.595 vs. 4.564, p = 0.040). Furthermore, the radius, diameter, eccentricity, and characteristics path length were greater (4.066 vs. 2.000, p = 0.009; 7.062 vs. 3.645, p = 0.029; 5.633 vs. 2.774, p = 0.013; 3.373 vs. 1.688, p = 0.004; respectively), whereas the average strength, global efficiency, local efficiency, mean clustering coefficient, transitivity, and small-worldness index were lower (5.595 vs. 10.831, p = 0.004; 0.350 vs. 0.642, p = 0.002; 0.531 vs. 1.724, p = 0.004; 0.304 vs. 0.624, p = 0.006; 0.456 vs. 0.935, p = 0.003; 0.913 vs. 0.993, p = 0.017; respectively), in patients with recurrent events than in those with a single event. The limbic covariance network shows significant alterations in patients with TGA, as well as differences between patients with recurrent events and those with a single event. These findings suggest that changes in the limbic covariance network could be related to the pathogenesis of TGA.

Relating the Outer-Independent Total Roman Domination Number with Some Classical Parameters of Graphs

For a given graph G without isolated vertex we consider a function f: V(G) rightarrow {0,1,2}. For every iin {0,1,2}, let V_i={vin V(G):; f(v)=i}. The function f is known to be an outer-independent total Roman dominating function for the graph G if it is satisfied that; (i) every vertex in V_0 is adjacent to at least one vertex in V_2; (ii) V_0 is an independent set; and (iii) the subgraph induced by V_1cup V_2 has no isolated vertex. The minimum possible weight omega (f)=sum _{vin V(G)}f(v) among all outer-independent total Roman dominating functions for G is called the outer-independent total Roman domination number of G. In this article we obtain new tight bounds for this parameter that improve some well-known results. Such bounds can also be seen as relationships between this parameter and several other classical parameters in graph theory like the domination, total domination, Roman domination, independence, and vertex cover numbers. In addition, we compute the outer-independent total Roman domination number of Sierpiński graphs, circulant graphs, and the Cartesian and direct products of complete graphs.

Transcriptome-Based Molecular Networks Uncovered Interplay Between Druggable Genes of CD8+ T Cells and Changes in Immune Cell Landscape in Patients With Pulmonary Tuberculosis.

BackgroundTuberculosis (TB) is a major infectious disease, where incomplete information about host genetics and immune responses is hindering the development of transformative therapies. This study characterized the immune cell landscape and blood transcriptomic profile of patients with pulmonary TB (PTB) to identify the potential therapeutic biomarkers.MethodsThe blood transcriptome profile of patients with PTB and controls were used for fractionating immune cell populations with the CIBERSORT algorithm and then to identify differentially expressed genes (DEGs) with R/Bioconductor packages. Later, systems biology investigations (such as semantic similarity, gene correlation, and graph theory parameters) were implemented to prioritize druggable genes contributing to the immune cell alterations in patients with TB. Finally, real time-PCR (RT-PCR) was used to confirm gene expression levels.ResultsPatients with PTB had higher levels of four immune subpopulations like CD8+ T cells (P = 1.9 × 10−8), natural killer (NK) cells resting (P = 6.3 × 10−5), monocytes (P = 6.4 × 10−6), and neutrophils (P = 1.6 × 10−7). The functional enrichment of 624 DEGs identified in the blood transcriptome of patients with PTB revealed major dysregulation of T cell-related ontologies and pathways (q ≤ 0.05). Of the 96 DEGs shared between transcriptome and immune cell types, 39 overlapped with TB meta-profiling genetic signatures, and their semantic similarity analysis with the remaining 57 genes, yielded 45 new candidate TB markers. This study identified 9 CD8+ T cell-associated genes (ITK, CD2, CD6, CD247, ZAP70, CD3D, SH2D1A, CD3E, and IL7R) as potential therapeutic targets of PTB by combining computational druggability and co-expression (r2 ≥ |0.7|) approaches.ConclusionThe changes in immune cell proportion and the downregulation of T cell-related genes may provide new insights in developing therapeutic compounds against chronic TB.